b J ■5fj_*We
16AU6#74
NflVflL RESEARCH
L
QUflRTERiy
DECEMBER 1973
VOL. 20, NO. 4
OFFICE OF NAVAL RESEARCH
NAVSO P1278
NAVAL RESEARCH LOGISTICS QUARTERLY
EDITORS
F. D. Rigby
Texas Tech. University
B. J. McDonald
Office of Naval Research
O. Morgenstern
New York University
S. M. Selig
Managing Editor
Office of Naval Research
Arlington, Va. 22217
ASSOCIATE EDITORS
R. Bellman, RAND Corporation
J. C. Busby, Jr., Captain, SC, USN (Retired)
W. W. Cooper, Carnegie Mellon University
J. G. Dean, Captain, SC, USN
G. Dyer, Vice Admiral, USN (Retired)
P. L. Folsom, Captain, USN (Retired)
M. A. Geisler, RAND Corporation
A. J. Hoffman, International Business
Machines Corporation
H. P. Jones, Commander, SC, USN (Retired)
S. Karlin, Stanford University
H. W. Kuhn, Princeton University
J. Laderman, Office of Naval Research
R. J. Lundegard, Office of Naval Research
W. H. Marlow, The George Washington University
R. E. McShane, Vice Admiral, USN (Retired)
W. F. Millson, Captain, SC, USN
H. D. Moore, Captain, SC, USN (Retired)
M. I. Rosenberg, Captain, USN (Retired)
D. Rosenblatt, National Bureau of Standards
J. V. Rosapepe, Commander, SC, USN (Retired)
T. L. Saaty, University of Pennsylvania
E. K. Scofield, Captain, SC, USN (Retired)
M. W. Shelly, University of Kansas
J. R. Simpson, Office of Naval Research
J. S. Skoczylas, Colonel, USMC
S. R. Smith, Naval Research Laboratory
H. Solomon, The George Washington University
I. Stakgold, Northwestern University
E. D. Stanley, Jr., Rear Admiral, USN (Retired)
C. Stein, Jr., Captain, SC, USN (Retired)
R. M. Thrall, Rice University
T. C. Varley, Office of Naval Research
J. F. Tynan, Commander, SC, USN (Retired)
J. D. Wilkes, Department of Defense
OASD(ISA)
The Naval Research Logistics Quarterly is devoted to the dissemination of scientific information in logistics and
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Information for Contributors is indicated on inside back cover.
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Permission has been granted to use the copyrighted material appearing in this publication.
GENERALIZED MULTICOMPONENT SYSTEMS UNDER
CANNIBALIZATION
Murray Hochberg
Brooklyn College (CUNY)
Brooklyn, N.Y.
ABSTRACT
Using the theory of Hirsch, Meisner, and Boll, we study the consequences of inter
changing parts within a generalized coherent structure. This procedure has been termed
"cannibalization." The theory of cannibalization is extended to the case where each compo
nent can operate at several levels of partial performance and a representation theorem is
derived, which expresses the state of a system as a function of the number of working parts
at each level. The stochastic theory of these systems is then investigated.
INTRODUCTION
Coherent structures and similar multicomponent systems have been studied extensively [1, 3, 4].
In particular, Hirsch, Meisner, and Boll [3] have studied the effect, of shifting operational parts from
one location to another, a procedure called "cannibalization." The author uses the theory of Hirsch,
Meisner, and Boll to study the consequences of interchanging parts within a generalized coherent
structure. The theory of cannibalization is extended to the case where each component can operate
at several levels of partial performance and the structure is permitted to take on several possible values
of performance. The main result is a representation theorem, which expresses the state of a system
as a function of the number of working parts at each level. The stochastic theory of these systems is
studied and a formula is derived for the probability distribution of the cannibalized structure function.
1. ALGEBRAIC THEORY
We introduce the abstract set A = {Ai, A 2 , • • ., A„}, whose points represent the loci of the
structure. We assume that at each moment of time each locus is in one of the following k possible
states:
(a) It is in the "best" operational state, in which case we associate to the locus the value ai, ai = 1.
(b) It assumes one of the £ — 2 "intermediate" operational values, {a.z, a 3 , . . ., aici} with
I = ai>a2>a3> • • ■ >afc_i>0.
vc) It assumes the value a*, ajt = 0, which signifies that the locus fails to contain an operational
part. Thus, at any fixed moment of time, the states of all the loci are described by a mapping
r:A» {a u <h, . . ., Ofc},
whose value at A*,
denotes the state of A*. We will denote the totality of the possible states of the loci by K", where
585
586 M. HOCHBERG
K n ={(zi,z 2 , . . ., Zn):zi = a h i = 1 , 2 , . . ., n, j=l,2, . . ., k}.
Each point veK" is called a locusvector state.
We shall assume that two parts 0\ and Q> can be interchanged if and only if the following two
conditions are satisfied:
(1.1) They are capable of functioning in precisely the same set of loci.
(1.2) When #i is installed and operating at a given level in a given locus k, its "contribution" to
the "level of performance" of the structure is precisely the same as that of Lh, when Ck is installed
and operating at the same given level in A. However, the contribution of a given part to the level of
performance of the structure may depend on the locus A. in which the part is installed.
We shall make the assumption that the level of performance of a structure at a given moment
of time is determined solely by the locations and states of all of its parts at that moment. Assumptions
(1.1) and (1.2) then imply that the level of performance of the structure is determined, at any given
instant, by the states of the loci. We assume that some measure of performance has been selected,
and we denote the set of possible performance levels by S = {O, 1, . . . , M}. We interpret the state
"0" as total failure and the state 'VW" as perfect performance, and we call each possible level a system
state.
We define a partial order in K n by setting y ^x if and only if (y); =£ {x)u i= 1, 2, . . ., n, where
(x)i denotes theith component of x.
We assume that we are given a structure function
<f>:K n ^S,
whose value at the vertex veK" denotes the state of the system when the loci are in the states described
by the vector v. A function <f>:K n ^> S is said to be a monotone structure function if
(a) v =£1/ implies that 4>(v) =£ <f>(v')
and
(b) <M0)=0, <f>(l)=M, where 0= (0,0, .. .,0) and 1 = (1, 1, . . ., 1).
If k = 2 and M — 1, a monotone structure function is called a coherent structure. Without further explicit
mention, all the structure functions considered in the sequel will be assumed to be monotone.
The set of parts satisfying (1.1) and (1.2) defines a part type. To make precise our assumption about
interchangeability, we introduce the set
T={yi, 72, • • ., 7n},
where yi, 72, • • ., 7v represent the part types in the structure. Let 2 A denote the collection of all
possible subsets of A We suppose that we are given a mapping
<? : r^2\
MULTICOMPONENT SYSTEMS CANNIBALIZATION 587
whose value Q(yi) represents the set of all loci in which parts of type ji are installed initially; then,
under our restrictions of interchangeability, interchanges are nermitted within Q(yi), but not between
a locus Ac(?(yf) and a locus k'4Q(yi). We postulate in addition that
(?(yi)*0, i=l,2, . . .,N,
(1.3)
Q(yi)nQ( yj ) = 4>, i£i*j
and
U Q(7i) = \.
J = 1
These properties express, respectively, the facts that each part type is associated with at least one
locus, no locus has more than one part type associated with it, and each locus has some part type
associated with it. Thus for each integer i, 1 =£ i =£ n, there is a unique./', say 7 = 8(1"), that satisfies
the relation ^i€Q(yj). The function 8 identifies the part type used at each locus.
Definition. Given a set A, the indicator function, I a, is defined by
1 ( \_fl> ifveA
Ia{v) \0, if*,
Definition. For each integer 7, 1 =s 7 «£ N, we set
Wj= (Wju Wj 2 , . . . ,U)j tk i).
where
w Jq :K*^>{0, 1, . . .,n}
is the map defined by
wjq(v)= 51 h(v\ = ag) (v) , g=l,2, . . .,k — 1.
{i:X ( «<?( y> )}
The map wjq can be interpreted as the number of operational parts of type yj operating at level a g ,
<7=1,2, . . ., jbl.
The following properties of Wj are immediate:
(a) 0< 2w jg (v)^\Q(yj)\
kl
2
9=1
(b) '*£u>j q (v) = \Q(y J )\,
9=1
if and only if all kieQ (yj) are operating at some positive level of performance.
588 M  HOCHBERG
(c) w jl (v) = \Q(y j )\
if and only if (v)i — 1 for all i such that ^teQ(yj).
Definition. Two vertices veK n and v eK n will be called wyequivalent, in symbols,
V ~ V
if and only if
Wj(v) = Wj{v').
Definition. The vertex v is said to be equivalent to the vertex v' , in symbols,
v ~ v' ,
if and only if for each integer j, 1 *£_/ =£ TV, we have
Wj
v~v'.
Wj
Clearly, "~" and "~" are equivalence relations, and we will denote the respective equivalence
classes by [v] Wj and [»]. The class [v] Wj consists of all locusvector states, v' , for which the number
of operational parts of type jj operating at level a,, i= 1,2,. . ., k — \, is the same for both v and v' .
The class [v] consists of all locus vector states, v' , for which the number of operational parts of type
yj, l^j^ N, operating at level a,, i=l, 2, . . . , k — 1, is the same for both v and v' . Since we have
postulated that only parts of the same type can be interchanged, and since an interchange clearly does
not affect the number of working parts of any part type at any level, the operation of interchanging
parts leads from a vertex v to a vertex v' in [»]. Guided by this we introduce the following definition:
Definition. A cannibalization is any transformation
T:K n ^K n
such that for all veK",
Tve [»].
We denote the class of all cannibalizations by 3~ .
Given a cannibalization T, we define the cannibalized structure function, <f> T , by
<b T (v) = (f>T(v)=<b(Tv).
A cannibalization T is said to be admissible if
(1.4) <j>t^<]>t> for all T 6^",
MULTICOMPONENT SYSTEMS CANNIBALIZATION 589
which expresses the fact that an admissible cannibalization is uniformly as good as any other can
nibalization. Plainly, T is admissible if and only if
4> T (v) = ma\(f)(v'), veK n .
Since there may be many points in [v] at which <£ assumes the value max <f>(v'), in general there
Vf[V\
are many admissible cannibalizations. We will denote the class of admissible can nibalizat ions by<^*.
It is clear that each Te^~* induces precisely the same cannibalized structure function, which we
denote by 4>*. Clearly, <f>* is constant on each equivalence class [v]. Moreover, <j> =£ <f>*, with equality
if and only if u ~ v implies <j)(u) = <f)(v).
Henceforth, we consider only admissible cannibalizations.
Definition. Let u be any point in [u] such that for all u'e[u],
We call such a point a maximum point of the restriction of(f> to [«]. This restriction is denoted by </>[u].
THEOREM 1.1: If <f> is a monotone structure function, then <f>* is also a monotone structure
function.
PROOF: Let u= (i u t 2 , . ■ . , in) and v= (ki, k 2 , . . . , k„) be two vertices in K n , such that
u *£ v. Let u be a maximum point of <f> \ [u]. We will construct a vertex ve [v] such that u =£ £. Since
ue [it],
" = (*«!» l Q 2 t • • • ' l 9 n ^'
where (i q , iq 2 , . . ., iq n ) is a permutation of (ii, 12, • ■ ,in) We now define
V= {k Ql , kq 2 , . . ., k q J.
ij^kj, ;'=1,2, . . . , n,
iQj^kQj, 7=1,2, . . .,n
u ^ i), with ve [v].
Since
we have
Thus,
Then
<l>*(u)=4>(u)^<l>(v)*z<l>*(v).
We now introduce concepts that enable us to measure the extent to which the performance of
the system depends individually on each part type.
590 M. HOCHBERG
Let II, : K n —> K" be the mapping denned by
LI, otherwise, ueK n .
The effect of Il< on a locusvector state is to transform it into one in which all the loci occupied by parts
other than type y% are operational in state 1 and in which states of the loci corresponding to part type
yt are left unchanged.
The following properties of If, are immediate:
(a) If, is nondecreasing; i.e., n,i; 5* v.
(1.5) (b) Iff is order preserving; i.e., if u 3= v, then IfjU 2= Fiji;.
(c) u ~ v implies that 11,u ~ X\iV.
Using the definition of a cannibalization T and (1.5c) we conclude that if u ~ v, then TTIjU ~ n,t;.
In particular, we have
7TI,i; ~ Ifij; ~ v.
(1.6)
For each integer i, 1 ^ i^ N, we define the structure function relative to i, <f>i, by
4>>=4>rL..
The function </>* describes how the structure would perform without cannibalization if an infinite
number of spares for all parts other than yj were available. Since If, is nondecreasing and order
preserving,
and, if </> is monotone, so is </>,.
We define the cannibalized structure function relative to i, <j>*, by
<^*(v) = ^*(n t t;) = < / > (7 , n i t;).
The function <b* is constant on each equivalence class [v]^; i.e., there is a function ]
/,:{0 f 1, . . ., IGCyi)!}* 1 ^
such that for all veK",
<f>*(v)=fiWi( V ).
1 We use the standard notation, A k = A x XA^X . . . XA k , where/4 ( = y4, i= 1, 2, . . ., k.
MULTICOMPONENT SYSTEMS CANNIBALIZATION 591
ltf
For if v ~~ i/, then by (15), Htv ~~ I1,V, and therefore
<l>*(v) = <f>*(Il i v) = 4>*(U i v') = <l>*(v').
THEOREM 1.2: If <f>* is a monotone structure function, then/; is nondecreasing in each variable.
PROOF: Let (xi, x 2 , . . ., x k \) < (yi, y%, . . ., y*i). Let u and v be two vertices such that
u>i(u)= (%u X2, . . ., Xki) and Wi(v)= (yi, y 2 , . . ., y*i). As in the proof of Theorem 1.1, we can
construct v'e [v] Wi such that u < v' . Since <f>* and n* are nondecreasing, we conclude that
fi(Xi,X 2 , . . ., X k i)=fiWi(u) = <l>f(u) ^<f>f(v')
= 4>*(v)=fim(v)=fi(yi,y 2 , . . ., y k i).
DEFINITION: A structure function </> is said to satisfy the minimum condition if
<b — min d>..
Intuitively, one can interpret the minimum condition as asserting that the value that the structure
function assumes at any vertex is determined by one particular part type, in the sense that if all other
part types were to be made fully operational, the value of the structure function remains unchanged.
The responsible part type may depend on the vertex and need not be unique.
THEOREM 1.3: If <f>= min <£,, then <£*= min <f>f.
THEOREM 1.4: Let <f> be any structure function that induces 0*. The relation
<f>*= min (f>f
holds if and only if to each maximum point v of <f> \ [v] there corresponds an integer io, depending on
v, such that n to t> is a maximum point of <£ [Hi v] and
<f>(v)=<l>(Ili v).
THEOREM 1.5: If </>*= min <f>f, then
lsjsAf J
min <f>7TIj= min SYljT.
We omit the proofs of Theorems 1.3, 1.4, and 1.5, since the proofs given in [3] can be easily ex
tended to the more general model considered here.
We are now in a position to derive an algebraic representation of the cannibalized structure func
tion </>*. We recall that
(f>f=fiWi,
592 M  HOCHBERG
where the function ft is nondecreasing in each variable. Let Ai, q , l=£js£/V, O^S^ssM+l, denote
the set of (k— l)tuples in the domain of/,, for which / takes a value at least as large as q. Since fi
is nondecreasing in each variable, we have
4oD4i2 • • • 24«24m+i. i=s;=s/v.
Since 4>f *£ M, we conclude that Ai,M+i = <i> Moreover, since <f>f(l, 1, . . ., 1) = M, we have Ai,M ¥" <}>.
Thus, Ai, q ^ <f> if q «? M.
We say that x is a minimal point of Ai, q if xeAi, q and there does not exist any ye4,, q , with y < x.
We define the set m(q) , =£ q =£ M, to be the set of minimal points of Ai, q . Moreover, we set
n,(Af+l){(K?(y,)+l f \Qy t )\ + l, . • ., \Q(yt)\+l)h
DEFINITION: Given two collections of vectors si and 99, we say that si 5= 99 (or 99 «£ si) if and
only if for every vector ye si there exists at least one vector ze99 such that y 3=z.
LEMMA 1.1: The relation ">" as defined above is a partial order on the set m(q).
PROOF: We must verify the three properties of a partial ordering:
(a) si^si.
(b) U si ^99 and 99 ^ si , then si = 99.
(c) If si ^ 39 and 39 ^ V , then si ^ «\
Properties (a) and (c) are obvious, and thus all we need to verify is property (b). Let ye si. Then since
si 3= 99, there exists ze39 such that y 3= z. Since 39 ^ si , there exists y'ej^ such that z 3* y'. Thus,
(1.7) y^z^y'.
Since y, y'eni(q), it is impossible that y > y' . Thus we have equality in (1.7), and y—z. By symmetry
it follows that si =99.
We define
Wi(v) 3= nt(q)
to mean
{wi(v)} ^m(q).
THEOREM 1.6: For all integers i, q, 1 *£ i =s/V, =£ q *£ M + l, and veK n ,
fiWi{v) 3= <?
if and only if
Wi(v) ^m(q).
PROOF: For q = M + l the theorem is true since w t {v) =£ (\Q(ji)\, (?(y.), • • ■ > <?(7i)) and
fiWi(v)^M. Assume now that fm(v) 3= q, =S g =£ M. Then Wi{v)eA i>q . \{ Wi{v)em{q), then surely
MULTICOMPONENT SYSTEMS CANNIBALIZATION 593
Wi(v) 5s rii(q). Hence, it will suffice to examine the case, Wi(v)*m(q). If u>i(v)*ni(q), there exists a
(k — l)tuple xen,i(q) such that wt(v) > x. Then Wi{v) 2= rii(q).
Now assume w(v) 3= rn(q). Since/, is nondecreasing and n t (q) C A i<g , we have that JiW t (v) 2* q.
Theorem 1.6 shows that the level of the cannibalized structure relative to i is at least as large as q
if and only if there are "at least" m(q) parts of type y, operating, where "at least" is to be interpreted
in the "2 s " sense.
THEOREM 1.7: Let q < q'. Then m(q) =£ m(q').
PROOF: First consider q < q' *£ M. As indicated previously, Ai,q>(ZAi, q . In particular, rn( q') QA i<q .
Let x' be an arbitrary (k — l)tuple in rii(q'). If for every x €n.i(q'), x' is also in nt(q), then m(q) =£ niq').
If there exists an x' enj(g'), x' erii(q), there must exist xen.i(q) with x < x . Then iu{q) =S rii{q').
For q' = M+ 1 the theorem is true since yeAi° q , =£ q s£ M, implies that all the components of y
are less than or equal to (?(y*).
We derive finally a useful algebraic representation of the structure function.
THEOREM 1.8: (Representation Theorem): If <f>*= min <fo*, then
M X M
**=xnw))=s^ { ^ (i))
k=l i=l fc=l '>
PROOF: Let t>€/£" be arbitrary and let <j)*(v) = k v . Then by hypothesis there exists an integer
io, 1 ^ io < N, such that
min <l>?(v) = <f>i*(v) =fi w io (v) = k v .
We first consider the case, k 5* k v + 1. Since </>* (f) 2= &», Theorem 1.6 imphes that Wi„(v) 2* «(„(£„).
Moreover, since <f> t *(v) <k v +l, it follows from Theorem 1.6 that
(1.8) w io (v)>n io (k v +l).
Since k 3= k v + l, we conclude from Theorem 1.7 that
(1.9) n io (k v +l)^n u (k).
Relations (1.8) and (1.9) imply that
w io (v) £n io (k).
Thus, for A: 2= k v + 1 ,
/{u>i o ( U )sn io (fc)}(f) = 0,
which implies that
/v
(1.10) ~\Hm>n>(k)h{v) = Q.
594 M HOCHBERG
Now suppose that &*£ k v . Since <£* = j™!",,^*' an d since <f)*(v) = k v , we have k v ^<f>*(v) for all
integers i, 1 *£ i =S N. Using Theorems 1.6 and 1.7, we conclude that
rii(k) ^ m(k v ) ^ wi(v) , l^i^N.
Thus for k*S k v , we have
(i.ii) riw«<*»(«o=i.
By combining (1.10) and (1.11),
fc=J i = l A = l i = l fr=fcu+l i = l
= * v + 0= *„ = <£*(!;).
Since, for any collection of sets {Ai}^ =l ,
n /«, i=i
the proof is complete.
2. STOCHASTIC THEORY
We now introduce into our model random variables that make it possible to study mathematically
the role of chance in the life history of a system subject to cannibalization. Our main goal is to deter
mine at a given moment of time the probability that the system is operating at level k, k — 0, 1, . . . , M.
We consider a system governed by a monotone structure function <f> and assume at time t =
there are s, spares available of part type yu i=l, 2, . . . , N. There are many possible service pol
icies prior to stockout of a given part type (i.e., before spares have been exhausted). The policy we
consider is the one in which a spare is installed only when a part has degenerated to the "totalfailure"
stage; i.e., its locus has operational value 0. After the supply of spares of a given part type has been
exhausted, a failure of that type is serviced by performing an admissible cannibalization. t For mathe
matical simplicity, we assume that cannibalizations and replacements of failed parts are performed
instantaneously.
tit should be noted that the results obtained in this paper remain valid if admissible cannibalizations are also performed
following each partial part failure, both before and after stockout.
MULTICOMPONENT SYSTEMS CANNIBALIZATION 595
We define the stochastic process t
where for each fixed t, F T) is a random variable whose possible values are a.j, 7 = 1, 2, . . ., k. We
interpret V^V (t) as the state of the part in locus Xi at time t if the cannibalization T is used. We set
<P*(t)=<(>*V T (t)
and
wj{t)= uii vm).
Clearly, <£>*(t) represents the state of the cannibalized system at time t, and WJ{t) represents the
random number of operating parts of type y, at time t.
Assuming that <f>* satisfies the minimum condition, it is easily shown from the representation
theorem that
M N
(2.D ^w = snW)M*)]
k=\ i=l
M
2
X J n {WJ(t)^n t (k)) .
As shown in [3], the probability distribution of 4>*(£) may depend upon the particular admissible
cannibalization used. In order to eliminate the dependence of the probability distribution of 4>*(t) on
7e«!F, we postulate the following:
(a) The failure rate at each instant of a given part type y,, i = l, 2, . . . , N, does not depend on the
particular locus in Q(yi) in which the part is installed, nor on the particular sequence of loci through
which it has passed.
(b) Parts operate independently; i.e., the lifetime of a given part is not related to the lifetimes of
any other parts.
Postulate (a) implies that the joint distribution of (WT(t), W r 2 (t), . . ., W T N (t)) doesn't depend on
the particular cannibalization TtST employed. Postulate (b) implies that the /V stochastic processes
{r,(o,ts*o}, {r 2 (o,*^o}, . . .,{w N {t),t^o)
are mutually independent; the superscript T is omitted because of postulate (a).
t Although the stochastic process {V T (t), t>0} and the other processes discussed in this section are not explicitly con
structed, it will be plain that in an appropriate model all the functions that we consider are measurable.
596 M. HOCHBERG
Hence, using (2.1), we conclude that the expected state of the cannibalized system at time t is
M N M N
(2.2) £[**(«)] = 2 II Eihw^m] =2 n p W<<) * *<*»•
fc=l i=l *=1 i=l
where P is the underlying probability measure defined on 2*".
Setting
and recalling that
m(l) < m(2) < ■ • • ^mW,
we conclude that
(2.3) / t >/,>...» /«.
Hence, it follows from (2.1) and (2.3) that
{**(*) ^}={i/*^;U{/j=i}.
Therefore,
(2.4) p{**(0 ;*./} = m=i}
=p(n {r,(t) >*(/)})
= ft W,(*) **(/)},
where the last step follows from postulate (b).
Thus, in order to evaluate the probability distribution and the expected value of <P*(0, it is neces
sary to compute P{Wi(i) 3= ra*(/)}» i = l> 2, . . .» N,j=\, 2, . . ., M. The remainder of this section
is devoted to this calculation. Our calculations will apply to an arbitrary, but fixed part type yi, and
accordingly we shall omit the subscript i from all the relevant expressions.
We begin with the postulate that the lifetimes of parts of a given type are identically distributed
random variables. Moreover, postulates (a) and (b) imply that the probability distribution of W(t) is
independent of the particular admissible cannibalization used and, indeed, of whether or not canni
balization is practiced. Thus to determine the probability distribution of W (i), it suffices to consider a
model in which at time t = 0, \Q\ parts of type y are installed and functioning at level 1, s spares are
available, and cannibalization is not practiced.
MULTICOMPONENT SYSTEMS CANNIBALIZATION 5^7
For simplicity, we treat the case in which there are only three possible operational states for each
locus: ai = 1, a 2 , and a 3 = 0, with < a 2 < 1. Thus, in order to use (2.2) and (2.4), we must evaluate
(25) P{M)=j, 6(0=*},
where W{t) = (fr(t), frit)), and fr(t) denotes the number of operational parts at level a v at time t.
We consider first the case when 5 = 0. Let O u 2 , . . ., 0q denote the \Q\ installed parts, and
let Z v (Oj) be the random variable representing the lifetime in state a„ of part O h v=l, 2,j=l, 2,
. . ., 0. Since the performances of parts in different loci are independent of each other, Z t (Oi),
Zi{0 2 ), . . ., Zi(0\ Q \) are independent, identically distributed random variables with a common
distribution function, say F. Similarly, Z 2 (O t ) , Z 2 (0 2 ), . . ., Z 2 {0\ Q \) have a common distribution
function, say G, and, by assumption,
Z l =(Z l (O l ),Z 1 (0 2 ), . . ., Z,(0«,))
and
Z2=(Z 2 (O l ),Z 2 (0 2 ), . . ., Z 2 (O w ))
are independent random vectors. Moreover, we assume that Z x {Oj) and Z 2 {0 } ) are independent ran
dom variables, j= 1, 2, . . ., \Q\.
We define A (t) , B(t) , and C{t) , respectively, by
(2.6) A(t) = P{Z l (O j ) + Z 2 (O j )<t},
(2.7) B(t) = P{Z 1 (O j ) < t, Z i (O j )+Z 2 (O j ) s* t},
and
(2.8) C(t) = P{Z l (O j )^t};
by assumption A(t), B(t), and C(t) do not depend onj. Clearly, A(t) is the probability that a part in
stalled at time t = will pass to state before time t. We call this event a complete failure. B(t) rep
resents the probability that a part installed at time t = will pass to state a 2 before time t and that it
will remain in state a 2 at least until time t. We call this event a partial failure. C(t) is the probability
that a part installed at time t=0 will remain in state 1 at least until time t. Plainly,
(2.9) A(t)+B(t) + C(t) = l.
Moreover,
(2.10)
A{t) = F*G{t)= P F(tx)dG(x),t^0.
598 M. HOCHBERG
Similarly,
(2.11) B{t) = [' dF(u) [" dG(v)
= F(t)F*G(t), t^O.
Clearly,
(2.12) C(t) = lF(t), t^O.
Using (2.10), (2.11) and (2.12), we conclude that for 5 = 0,
101
(2.13) P{^( t )=j, &(f) = *} = (/, *, \Q\jk) [lF(t)]J[F(t)
X ' F*G(t)] k [F*G{t)]WJ k ,t^0.
Suppose next that s > 0. We shall not treat this case in full generality. Instead, we assume that
the random variable Z„(Oj), v= 1, 2, j = 1, 2, . . ., 0, has an exponential distribution with param
eter k v > 0; thus,
f 1 — p~ k \ x x ^
and
G(x) = P{Z 2 (Oj) <*} = { 1
1eV, * 5*
a; < 0.
Let Z v(0\q\+ic) , P= 1, 2, be a random variable denoting the lifetime in state a„ of the A;th spare to be
installed, jfe— 1, 2, . . . , s. Since the spares and the originally installed parts are of the same type,
our assumptions imply that for v=l, 2, Z„(0i), Z„(0 2 ), . • .,Z„(0\q\), . . ., Z v (0\ Q \ +g ) are inde
pendent and exponentially distributed with parameter \„.
Under these assumptions, (2.10), (2.11) and (2.12) become, respectively,
(2.14)
A{t)=[ [le x i «*)]\ 2 e x **dx
Jo
= le x "r 4" > X2 'e x "], t 2* 0,
A.1 A.2
(2.15) fi(t) = [ \ x e^du I" \ 2 e x ™dv
A1
Xi — A. 5
[e x "e Xl< ],. f 5* 0,
MULTICOMPONENT SYSTEMS CANNIBALIZATION 599
and
(2.16) C(t) = 1  [1  e x «] = e x ", t ^ 0.
It is convenient to consider next the hypothetical situation in which there are available an infinite
number of spares of the part type under consideration. In this case, whenever a part fails completely,
it is immediately replaced by a new part, which is in state 1. Thus, at any moment of time, each of the
 Q  loci are either in state 1 or state a 2 .
Choose any locus and let /u.„(t) denote the probability that, at time t, the component in use in
that particular locus is in state a„, v = 1,2. Since the lifetime in state 1 is exponential with parameter
Ai, it follows from standard arguments that the conditional probability that at time t + h a part in a
given locus is in state a 2 given that at time t it was in state 1, is kyh + o{h). Similarly, the conditional
probability that at time t + h a part in a given locus is in state 1 given that at time t the part in this
locus was in state a 2 is \ 2 /i + o(h). Moreover, the probability that more than one passage occurs in a
given locus in the time interval (t, t + h) is o(h), and the process describing the state of the locus at
each moment of time is Markovian. The preceding observations imply that
yn(t + h) = [1 — \i/i]/u.i(0 + A 2 /i/li 2 (0 + o(h),
or
^ =  Ai/MO + ^2jM*) H jp
Letting h —* 0, and using the fact that
fl 2 (t) = 1  /Xi(t),
we obtain the differential equation
(2.17) fi' 1 {t) + (\i + \ 2 )iJii(t)=\2.
The initial condition for this first order linear differential equation is
(2.18) /*i(0) = l,
which signifies that at time* = Othe locus is in state 1. The solution of (2.17) under the initial condition
(2.18) is easily seen to be 3
3 Expressions (2.19) and (2.20) are also derived in [2] by the use of various theorems in renewal theory. Our derivation uses
first principles.
600 M. HOCHBERG
Thus,
Since the lifetimes of all parts in a given state are equidistributed, we conclude that in the case of
an infinite number of spares,
(2.21) P{Ut)=j,^(t)=\Q\j}=^ Q . l )[ f Ji l (t)]n^(t)]WJ.
For large values off, (2.21) can be approximated by using the limiting relations
J_
lim ix v {t ) =  — !L j, v=\,2.
t—*t» ,
These asymptotic formulae are not surprising, since the mean of the lifetime of a part in state v is l/A.*.
We consider now the case when s < °°, which in turn falls naturally into the following two subcases:
(a) A<  Q  Jin (2.5)
and
(b) A:=<?jin(2.5).
Case (a) can occur only after stockout, while case (b) can occur either prior to or after stockout.
Letting r)(t) denote the number of complete failures up to time t in all of the loci combined, it is
obvious that the time of stockout Y is given by
y=inf {t : tj(t) 3* s}.
We wish to find the distribution function of Y. For y 3 s 0,F s (y) denotes the probability that the time of
occurrence of the 5th complete failure will precede y. Equivalently, 1 — F s (y) equals the probability
that the number of complete failures occurring up to time y is less than s. Therefore,
(222) l_ jFs(y) = /> o(y)+ p i(y) + . . . + /V,(y), i
where P„{y) denotes the probability that the total number of complete passages occurring in all the
loci up to time y is equal to v, v — 0, 1,. . .,s — 1.
MULTICOMPONENT SYSTEMS CANNIBALIZATION 601
In order to use formula (2.22) it is necessary to evaluate P v (y). To this end we first determine the
density function, h, of the random variable Zi(Oj) +Z2(Oj). Clearly,
h(t)=f*g(t),
where
and
n) to, t<
f x 2 <
"1 o,
\ 2 e k2t , t St
Thus,
(223) h(t)= (' At~y)g(y)dy
[e k 2 t ~e x i t ], t2*0.
Xi X2
LEMMA 2.1: Let h* m denote the mfold convolution of h and set
(2.24) H,„(t)= [' [lA(tz)]h* m (z)dz, m=l,2,...,s,
where the function A is defined in (2.14). Then the probabilities P„(t ) , v = 0, 1, . . .,5, satisfy the rela
tion
(2.25) P„(t) = £> ( \Q\ ) [lA(t)Yo[Hi(t)]«i[H 2 (t)] k 2 . . . [H v (t)]*K
PROOF: For v complete failures in all of the loci combined to occur there must exist integers
kj, =Sj =£ v, such that there are precisely j failures in each of kj loci, where
and
2 J k J = "'
k =\Q\^kj.
602 M HOCHBERG
We consider first the case where an infinite number of spares are available. Let L , r denote the
lifetime (i.e., time to a complete failure) of the part originally installed in locus A. r and let L m , r denote
the lifetime of the rmh replacement in locus A. r , m= 1, 2, 3, . . . . We set
X'n,r = i'0,r + ^l,r+ • • • +^ml,r
and
H m ,r{t)=P{Xm,r<t, \m ,r+ L m ,r ^ *} .
It is clear that the density of Xm.r is independent of r and is equal to the mfold convolution of A. More
over, Xm,r and L m , r are independent random variables. Hence
H m ,r(t)=\  h* m {z)h{w)dwdz
JO Jtz
= I [\A{tz)]h* m {z)dz = H m {t).
Clearly, H,„,r(t) represents the probability of exactly m complete failures occurring in locus Xr before
time t.
If we denote by M[ x) the event "exactly v complete failures in all the loci combined prior to time
V , in the infinite model (i.e., when there are an infinite number of spares), it follows that
P(MM)= , ? ( \Q\ ) [lA{t)]'»[H l (t)]'»[H i (t)]» . . . [HAt)] k »=PA0.
{*j:S>*j="( V Ad, k u . . ., k v I
)'«
In a similar fashion, it can be easily shown that the Lemma also holds for the finite model. Q.E.D.
We set
D(t)=P{Z 2 (Oj)^t}
and
E(t)=P{Z 2 (Oj)<t};
it is plain that the functions D and E do not depend on j . Clearly,
feV, t 3*
(2.26) D(t) =[ Qi t<Q
and
\« ' ;<o.
we,
MULTICOMPONENT SYSTEMS CANNIBALIZATION 603
We now consider case (a), k < \Q\ — j in (2.5). Let A v , v= 1,2, . . ., \Q\, denote the event that
v loci are in state 1 and \Q\ — v loci are in state a 2 at time of stockout, y. Plainly,
l«l
P{A, U^U...U A\ Q \) = 2 P(A„) = 1.
We note that v cannot be zero, since at stockout we install the last working part and it is in state 1.
Let Bm denote the event that the part which fails at time of stockout is in locus X m , m= 1, 2, . . ., \Q\.
It is clear from our assumptions that
that
and by symmetry that
Then
P(B m D B„) = 0, for m * n,
\Q\
2 P(B m ) = 1,
It is plain that
P(fl m )=^, m=l,2, . . ., \Q\.
P{A V ) = J P(A v \B m )P{B m )=r$r f) P(A„\B m ).
m =l IVI m=l
P{A v \B m )  ('^_ j 1 ) [My)Y l \.My)Y Ql ,
so that
= 1,2,..., <?.
(2.28) P{A V ) = {W_ X X ) [/*i(y)]" 1 [^(y)] tc ^. " = 1. 2
We set
We first evaluate
P{Rj, k (t)\Y=y}, k<\Q\j, t>y.
At time of stockout there exists an integer v, l«c« \Q\, such that v loci are in state 1 and \Q\ — v
loci are in state a 2 . We denote the collection of v loci which are in state 1 at time y by d and the collec
604 M  HOCHBERG
tion of \Q\ — v loci which are in state a 2 at time y by &. The occurrence of the event Rj,kU) specifies
that at time t> y, j loci are in state 1 and k loci are in state a 2 , and this can occur only in the following
way: j loci from s& remain in state 1, q loci from stf fail partially, v—j — q loci from s^ fail completely,
k — q loci from & remain in state a 2 and \Q\ — v — k + q loci from & fail to state 0, where 1 < v< \Q\,
j^p, O^q^ min (k, v—j) and ^ \Q\ — v — k + q. Recalling that the individual loci are independent
of each other, we conclude that
(2.29) P(R jtk ( t )\Y=y) = f i J ( V _._)[C(ty)V[B(ty)]"[A(ty)y^
\Ji ?! " J qi
v=\ <J =
( , 4!^ F )w*y)]* , tf(«y)] w  r ** , (^!r i 1 )[Mi(y)] , ' I D«.(y)] w  F .
We let K,(y) denote the right hand side of (2.29). Then, for k < \Q\ j
(2.30) P{ii(t)=j, &(*) = *} = f'x t (y)rfF.(y).
Jo
We next consider case (b), A; = \Q\ — j in (2.5). Plainly,
Pfoit) = j, 6(0 = l(?l  ;}  Wj.mjWI* > 0W > t) + P{Rj,wj(t)\Y < t}P{Y < t}.
Prior to stockout our model is the probabilistic prototype of the case of infinite spares, and thus from
(2.21) we conclude that
(2.3D p{Rj,\Q\j(t)\y^t} = ^[ f i 1 (t)V[Mt)V Ql  i .
Since k = \Q\ — j , we have q— v — j and (2.29) becomes
(2.32) P{R j ,\Q\j(t)\Y=y<t}
= 2 (J) [C(ty)V[B(ty)ynD(ty)V^{j Q J~ l 1 ^ [>i»(y)]»[/i.(y)]l«l'.
We denote the right hand side of (2.32) by L t (y). Then
(233) P{Rj,\Q\j(t)\Y < t} = f^ L t (y)dF s (y).
Using (2.31) and (2.33), we conclude that
(2.34)
p{M)=i,M)=\Q\jy=(Wypi{t)y'fam
MULTICOMPONENT SYSTEMS CANNIBALIZATION 605
In conclusion, we note that the righthand side of (2.2) and (2.4) can be evaluated using (2.13)
for part types y, having Si = and (2.30) and (2.34) for part types with st > and exponential lifetimes in
both states.
One should note that all the calculations in this section can be generalized to the case of A; levels of
performance with exponential lives in each state.
REFERENCES
[1] Birnbaum, Z. W., J. D. Esary, and S. C. Saunders, "Multicomponent Systems and Structures and
Their Reliability," Technometrics, 3, 5577 (1961).
[2] Cox, D. R., Renewal Theory (Methuen, London, 1962).
[3] Hirsch, W. M., M. Meisner, and C. Boll, "Cannibalization in Multicomponent Systems and the
Theory of Reliability," Nav. Res. Log. Quart. 15, 331359 (1968).
[4] Simon, R. M., "Optimal Cannibalization Policies for Multicomponent Systems," SIAM J. Appl.
Math 79, 700711 (1970).
A BAYESIAN APPROACH TO DEMAND ESTIMATION AND
INVENTORY PROVISIONING*
George F. Brown, Jr. and Warren F. Rogers
Department of Management
Naval War College
Newport, Rhode Island
ABSTRACT
This article addresses the problem of explicitly taking into account uncertainty about
the demand for spare parts in making inventory procurement and stockage decisions. The
model described provides for a unified treatment of the closely related problems of statistical
estimation of demand and resource allocation within the inventory system, and leads to an
easily implemented, efficient method of determining requirements for spare parts both in
the early provisioning phase and in later periods of operations when demand data have
accumulated.
Analyses of the model's theoretical foundations and of sample outcomes "f the model
based upon data on parts intended for use in the F14 lead to conclusions of great impor
tance to both support planners and operations planners.
Finally, of particular significance is the ability afforded the planner by this model to
quantify the impact on inventory system costs of varying levels of system reliability or
management uncertainty as to projected system performance. This will provide an economic
basis for analysis of such alternatives as early deployment, operational testing, and equip
ment redesign.
I. INTRODUCTION AND SUMMARY
Numerous studiest have demonstrated that the demand for aircraft spare parts is typically un
corrected with identifiable program factors. In the absence of such deterministic predictors, statistical
estimation procedures provide the best alternative means of estimating future requirements. Statistical
estimation consists of specifying the probability distribution of demand which, in some sense, best
explains the available data or, in the absence of data, best reflects the prior beliefs of the designer and
the experience of the inventory manager. Having specified the probability distribution, it is necessary
to determine the optimal inventory level as a function of the associated costs and budget constraints.
Typically the related problems of estimation and resource allocation are treated separately.** In
simple inventory problems this is probably justified. However, when planning support of an extremely
* Work conducted under contract N0001468AO091
tSee, for example. Denicoff, M. and Haber, S., "A Study of Usage and Program Relationships for Aviation Repair Parts,"
The George Washing i University, Logistics Research Project, Serial T140/62 (7 Aug. 1962). The probability model we devel
oped here is proposi 1 in th.s reference and many others on empirical grounds. The fact that small correlations may be expected
from data realized frt. n ' • jrocess does not appear to have been noted before.
**An exception is Zui !*_ , S., "A TwoEchelon MultiStation Inventory Model for Navy Applications,*' The George Washington
University, Logistics Research Project, Technical Memorandum, Serial TM15175 (31 July 1968), Zacks' approach is also
Bayesian and uses the same probability model as that developed here.
607
608 G  F  BROWN, JR. AND W. F. ROGERS
complex weapons system, with very great numbers of parts of widely varying cost and uncertain per
formance, a unified treatment of these problems is essential. In short, the objective must be to specify
the optimal inventory decision when system performance may be projected with only limited assurance.
This paper describes such a procedure for determining optimal inventory levels for aircraft
spare parts. The procedure may be used before demand data have been generated by incorporating
estimates developed at provisioning, and it provides for progressive updating of estimates as data
become available. The model is simple to apply, extremely efficient, and requires only existing data
sources. It is based on a few intuitive assumptions which have been repeatedly demonstrated to
correspond closely to data on existing systems. The model may therefore be used with confidence,
not only to determine inventory requirements, but perhaps more importantly to evaluate budgetary
and operational implications of support policies.
An application of these procedures to a number of parts currently being provisioned for the
F14 is described. From this application and from theoretical consideration, a number of very im
portant results are derived. The most important of these is that a spare parts inventory adequate to
assure high system reliability early in system life will be very costly and extremely wasteful. The
inventory required will be large but very little of it will actually be used. As data accumulate, however,
it will be possible to design inventories to provide equal reliability assurance at greatly reduced cost.
Thus, unless there are vital, overriding, operational requirements, the most desirable course of action
is to accept low reliability in the early life of the system; this means procuring parts as needed until
sufficient data have been accumulated to permit more economical inventory design.
Further significant results are summarized in the following paragraphs and are discussed in the
remaining sections of the report.
1. The model described in this paper provides for a unified treatment of the closely related prob
lems of statistical estimation of demand and resource allocation within the inventory system, which
are typically treated separately. A frequent criticism of theoretical inventory models is that they do
not reflect the uncertainty about the parameters which are inputs to the model — in particular, the
probability distribution of demand. The procedure described here explicitly introduces such un
certainties into the inventory decision process.
2. Uncertainty about demand distributions can result from a number of factors. At the time of
initial provisioning, estimates may be quite tentative due to the lack of any operational data on which
to base them. Further, a system designed to operate worldwide, in a host of unpredictable environ
ments, with a variety of maintenance procedures and skill levels supporting it, being employed in
widely varying missions, can be expected to have not one, but many, rates of demand. Both forms of
uncertainty are relevant to the inventory decision and are incorporated into the model described in
this report. Furthermore, these two types of uncertainty imply different requirements for inventory
support.
3. The model developed in this report enables the inventory manager to incorporate all of his
particular knowledge about a deployment into the optimal decision process. Peculiarities about a
particular deployment or a squadron's maintenance practices, as well as the size of the squadron and
the projected flying hour program, can be reflected in the inputs to the model.
4. The effect of uncertainty about the demand rate is to increase the variance of the probability
BAYESIAN DEMAND ESTIMATION 609
distribution of demands. In turn, this high variance typically implies higher required levels of stockage,
more frequent reordering, and, in general, higher costs of supporting the weapons system. This
high variance and associated high support cost have been frequently reported in studies of Naval
inventory systems. However, little guidance has been provided about what the Navy can do about
these problems. Our model suggests a number of management procedures which can be employed
to solve these problems beyond the usual suggestion that the equipment be redesigned so as to be made
more reliable. In fact, we demonstrate, in some cases, that a reduction in uncertainty can be of more
value than an equivalent increase in reliability. First, extensive operational testing can be under
taken to gather data which will lead to more certainty about demand rates. Planning to deploy ex
tensively an untested weapons system and to support it for wartime usage will require high levels of
inventory support. Furthermore, across parts, the higher the level of uncertainty, the greater will be
the percentage of this inventory which will go unused. However, it is impossible, a priori, to tell exactly
which parts will be used, so that extensive support across all such parts is required. Secondly, greater
standardization of maintenance facilities and practices will reduce the variance in this demand and thus
lead to lower inventory system costs. Finally, the ability of the inventory system manager to incorporate
information peculiar to a particular squadron and deployment can reduce the variance in demand that
the inventory system must protect against.
5. Numerous empirical studies of demand data have concluded that the observed pattern of
demands over time correspond well with the realizations of a compound Poisson process. The explana
tions advanced to support this conjecture have largely been unsatisfying. The model developed in this
paper, which follows from a few relatively mild assumptions, leads to one member of the compound
Poisson family — the Negative Binomial distribution. Thus the results of this paper are supported by
a wide body of previous empirical research.
6. A second major conclusion of previous empirical research has been that, with few exceptions,
demands for spare parts are uncorrelated with program factors such as flight hours. The model devel
oped in this report suggests that flight hours do enter into the determination of spare parts demands,
but in a very complex and distinctly nonlinear way. We show that, in fact, the theoretical model devel
oped here predicts the finding of a lack of correlation between flight hours and demand. The optimal
inventory decisions generated in the model involve a highly complex interaction among the param
eters of the demand distribution, relevant costs, and flight hours. Predictions of demand based upon
simple linear relations between demands and flight hours are overly naive and are based upon a faulty
premise.
Many of the mathematical results in this paper are well known. They are reproduced here both
for completeness and because their implications for support policy are extremely important and have
not been fully explored in the past.
The implementation of the procedures described in this paper should present little difficulty to
managers of the Navy's inventory systems. All of the procedures employed in the analysis, including
hose for determining optimal inventory decisions and for incorporating new demand information
as it becomes available, have been programmed and require only a few seconds of processing time.
The decision rules have been shown to be of a particularly simple form and thus can be used by man
agers of deployed squadrons.
610 G. F. BROWN, JR. AND W. F. ROGERS
II. A MODEL OF SPARE PARTS DEMAND
The Probability Model
Inventory decisions in Navy Supply are typically based on point estimates of demand. When
demands are subject to random variation, procedures based on point estimates will typically lead to
poor decisions. An optimal inventory decision model must consider the full range of possible realiza
tions of the random process which generates demands and their associated probabilities; the inventory
model described in section IV does this.* In this section, we derive a probability model of demands
which coincides well with empirical studies of demand data and is suitable for input in the inventory
model.
Numerous empirical studies of demand data have been conducted.! Three conclusions emerge:
a. With very few exceptions, demands for spare parts are uncorrelated with program factors
such as flying hours.
b. The Poisson distribution provides an adequate description of demands for parts exhibiting
low demand rates.
c. The variance of demands for high usage parts over time is typically very large compared to
their mean.
The latter observation has led to rejection of a simple Poisson model of the demand process for high
usage rate parts since the Poisson distribution has identical mean and variance.
Several conjectures have been offered to explain this behavior and to justify the choice of one
member of the compound Poisson family of distributions.** We have found these explanations un
satisfying either because they fail to correspond to operational experience or because the models
they were advanced to support would be inappropriate if in fact the explanations were valid. Instead
we show that one member of the compound Poisson family, the Negative Binomial distribution, follows
logically from some rather mild assumptions and some practical constraints imposed by the nature of
the estimation problem.
First we assume that demands for parts in nonoverlapping time intervals are statistically inde
pendent. It is easily shown (cf. Feller [4]) that any distribution on the integers which satisfies this
assumption is a member of the compound Poisson family.
Next we will assume that we may describe the uncertainty which exists about the anticipated
rate of failures, \, by assigning to it a probability distribution which summarizes designers', manu
facturers', and support managers' best "guesses" as to the values of mean time between failures which
may be realized when the equipment in question is placed in operation. The treatment of demand rate
as a random variable may at first appear strange to those unacquainted with Bayesian methods. Justifica
*A further treatment of the theoretical basis for this model is contained in Brown, George F., Jr., Corcoran, T. M., and
Lloyd, R. M., "Inventory Models with Forecasting and Dependent Demand," Management Science (Mar. 1971), and "A Dy
namic Inventory Model with Delivery Lag and Repair," Center for Naval Analyses, Professional Paper 3 (1969).
tFor example, see Fawcett, W. M. and Gilbert, R. D., "Characteristics of Demand Distributions for Aircraft Spare Parts,"
General Dynamics Fort Worth Division Report ERRFW512 (Nov. 1966). Also see Youngs, J. W. T., Geisler, M. A., and Brown,
B. B., "The Prediction of Demand for Aircraft Spare Parts Using the Method of Conditional Probabilities," RAND Corporation
Report RM1413 (Jan. 1955).
**For example, see Feeney, G. J. and Sherbrooke, C. C, "The (s — 1, S) Inventory Policy under Compound Poisson De
mand," RAND Corporation Memorandum RM4176PR (Mar. 1966). The authors offer four conjectures to explain the high
variability observed for recoverable item demand.
BAYESIAN DEMAND ESTIMATION 61 1
tion of this procedure is treated extensively in Raiffa and Schlaifer [5J and DeGroot [3 J. In this particular
application, however, it is intuitive that the underlying mean rate of failures which will be experienced
when an equipment is employed in the Fleet should be expected to vary randomly with varying and
unpredictable environments. We will refer to the distribution of k as the prior distribution.
For any given realization of failure rate per unit of time, say A, we will assume that the probability
of observing more than one demand in any very small increment of time is itself vanishingly small.
With this last assumption and the assumption of independence between nonoverlapping time inter
vals we may conclude* that the conditional probability of observing k failures in any time increment
t, given that the rate k holds is given by:
(1) W]=^f
If we denote our prior distribution on k by F(k), then the unconditional probability of observing k
failures in time t is:
(2) P(k)= r P(k\k)dF(k
Jo
(kt) k e xt
/;
k\
dF(k).
To this point, we have considered estimates of the distribution of demands based solely on prior
considerations; that is, before demand data have been generated. Naturally, as demand data accumu
late, we would wish to modify our prior beliefs about the mean demand rate to reflect this additional
information. This is accomplished by an application of Bayes rule as follows. Let/(\)= — jr — be
the prior density of k and suppose that in each of n time periods, £;, we have observed x, demands,
where i= 1, 2, . . . n. Then the conditional density of k, given the observations, is
(3) f(k\ Xl , . . .,*») = 7^ •
We will refer to the conditional distribution of k given the observations as the posterior distribution
of\.
With the additional information about demand rate summarized by the posterior distribution, the
unconditional distribution of demands in Equation (2) now becomes
(4) P[k] = l k\ /(X *" * • ■' Xn)dk 
*For a rigorous statement of the postulates leading to this distribution, see Feller (4).
612 G  F BROWN, JR. AND W. F. ROGERS
Choosing the Prior Distribution
To determine a suitable prior distribution F(\) , Raiffa and Schlaifer [5] established the following
desiderata:
"1. F should be analytically tractable in three respects: (a) it should be reasonably easy to deter
mine the posterior distribution resulting from a given prior distribution and a given sample;
(b) it should be possible to express, in convenient form, the expectations of some simple utility
functions with respect to any member of F; (c)F should be closed in the sense that if the prior
is a member of F, the posterior will also be a member of F.
2. F should be rich, so that there will exist a member of F capable of expressing the decision
maker's prior information and beliefs:
3. F should be parametrizable in a manner which can be readily interpreted, so that it will be
easy to verify that the chosen member of the family is really in close agreement with the
decision maker's prior judgments about 6 and not a mere artifact agreeing with one or two
quantitative summarizations of these judgments."
It is of particular importance in this application that the criterion 1(c) apply. If we choose a prior
distribution for which it did not, then the posterior distribution realized after each period of data collec
tion would have an algebraic form differing from that of the preceding stage. Thus, extensive repro
gramming would be required at each stage thereby effectively limiting the practical usefulness of the
procedure. We therefore choose a family of distributions which satisfies 1(c) and examine its other
properties.
A random variable A is said to be distributed as the two parameter Gamma distribution with
parameters a and B, denoted G a ,0, if its density is
(5) /(\)=p^\«ie*\
1 (a)
If the parameter X in the Poisson density given in Equation (5) has a prior distribution, G a ,/3, and
if Xi, i=l,2, . . . n, are n independent samples from that Poisson process, then the posterior dis
tribution of k is again a Gamma distribution with revised parameters
a' = a+^Xi, B' = B + n.
i=l
Thus a Gamma prior satisfies criterion 1(c) and coincidentally 1(a).
For this application the utility function is defined implicitly by the inventory program and thus
criterion 1(b) reduces to the requirement that the unconditional distribution of demands be computa
tionally tractable. From Equations (2) and (5) we derive the unconditional distribution of demands as
f co \ k e~ x B"
(6) P(k)=\ , , J] x ^"'e^dX
Jo
k\ r{a)
_( oc + kl\/ B \ a / 1 y
V k A/3+1/ \B+l) '
BAYESIAN DEMAND ESTIMATION 613
the Negative Binomial distribution with parameters a and — —  . A simple recurrence relation which
simphfies computation of the probabilities is given in section IV.
The Gamma family provides an extremely wide range of shapes, amply satisfying the second major
criterion.
The final criterion deserves more extended consideration. The Gamma distribution is completely
characterized by its mean and variance or by the mode and variance. The expected value (the mean)
and the most likely value (the mode) of the rate of demands are probably meaningful concepts to an
inventory manager or provisioner. It is doubtful, however, that variance is an equally meaningful
concept and that prior estimates of it would really reflect their prior beliefs as to likely system per
formance. This question is treated in more detail in INS Study 37 [2].
It is of interest to note that while the distribution in Equation (6) is compound Poisson, the random
process over the time parameter t is not. In fact, a nondegenerate mixture of Poisson processes cannot
yield a compound Poisson process. The distribution in Equation (5) is, in fact, infinitely divisible in the
parameter a not in t. 7
III. SOME IMPORTANT IMPLICATIONS OF THE PROBABILITY MODEL
Deployment of New Weapons Systems
The model we have described has great intuitive appeal in that its development follows from a
relatively few, mild assumptions, all of which appear consistent with operational experience. In addi
tion there is strong (and plentiful) empirical evidence that the model accurately reflects real world
experience. Predictions based on this model therefore merit serious consideration, particularly in view
of their implications for wartime contingency planning.
The demand distribution given in Equation (7) has mean and variance
E(*) = f,
v„ W *fc±4.
The mean and variance of he prior distribution given in Equation (5) are
Var(X) = ^
A large prior variance which implies a large uncertainty about X is thus reflected in a large uncondi
tional variance of demand. In addition the variance of the demand distribution increases quadratically
7 We are grateful to Dr. Joseph Bram, who called this point to our attention.
614
G. F. BROWN, JR. AND W. F. ROGERS
in flying hours t. The immediate implication of increasing variance is that the probabilities of large
demands also increase. To provide desired system reliability it is then necessary to procure larger
inventories. But for fixed mean demand in the probability that no demands will in fact occur* Thus
the likelihood that expenditures will be wasted also becomes large. Of course across parts, it is im
possible to tell with certainty which will be required and which will not.
Variance of demand is controlled by several factors. First there is the reliability of the system,
(X OL
reflected in the prior mean, — . Then there is the variance of the prior, — , which reflects the state of
uncertainty about the current estimate of demand rate. Finally, there is the projected flying hour rate.
One conclusion is immediate. A new weapons system, incorporating "state of the art" equipment,
whose performance may be projected only with great uncertainty, will require a large inventory of
spare parts to ensure acceptable reliability. If, in addition, it is intended that the system be capable
of sustaining an intensive wartime flying program, then the inventory must be expanded many times
over. In fact, the sample calculations given in section IV indicate that even with the penalty cost fixed
at the peacetime rate, which is no doubt unrealistically low, the war reserve inventory necessary to
ensure high reliability in the absence of resupply would far exceed the levels normally maintained.
An inventory policy designed to provide for wartime employment early in the life of the system
would not only be costly, but also extremely wasteful. It is important to realize the distinction between
the planned inventory necessary to assure readiness and the usage which will actually be generated
by the random process used in planning. The inventory must be designed to guard against demands
whenever there is significant probability that they will occur. The demand actually realized will reflect
the fact that there is also significant probability that a specific part will experience few or even zero
demands.
The alternative is to defer some procurement decisions until the acquisition of demand data permits
more reliable prediction of demand rates. As noted in section II, the posterior Gamma distribution of
demand rate after n realizations of the process yielding demands Xi, i = 1, 2, . . ., n, has parameters
a + JT Xi, /3 + n.
Then the posterior unconditional distribution of demands has variance
Var (k) =
(a + j? *«) *(/3 + n + t )
(/3 + n) 2
and thus the posterior variance decreases roughly as  . It follows that, in addition to allowing manage
n
ment to isolate those equipments whose realized reliability will dictate redesign action, deferral of
*The mean demand per flying hour is always less than one, so that for a fixed mean the increased mass at large values
must be "balanced" by an increase at zero.
BAYESIAN DEMAND ESTIMATION 615
major commitment of resources enables us to design future inventories providing the desired level of
readiness assurance, but at a greatly reduced cost.
Deferral of procurement, of course, implies acceptance of a reduced state of readiness in the early
stages of the program so that enhanced readiness and the ability to respond to contingencies can be
realized in later stages at acceptable cost. If, however, an initial high state of readiness and ability
to respond to contingencies is deemed imperative, then the inventory should be planned realistically
in the full realization that it will involve very great cost and potential waste.
Demand to Flying Hour Correlation
It has been noted earlier that estimates of the correlation between demand (or failures) and flying
hours based on observed data are typically small. We now demonstrate that this should, in fact, be
the expected outcome from data generated by our probability model.
Treating flying hours t as a random variable, we calculate the population correlation between
t and k, the number of failures, as follows.
The covariance of t and k may be written
Cov {t,k)=E{t(kE{k)))
= E{tk)E{t)E{k).
E{k)=E(E{k)\t)
Then
"&•)
*w.
E{tk)=E{E{tk\t))
=*($<)
= jE(t*),
whence
Cov (t, *)=Var («).
Since
Var (k)=E(k*)E 2 (k)
616 G. F. BROWN, JR. AND W. F. ROGERS
and
E(k 2 ) = E(Ek 2 \t))
=*(^MS<) 2 )
= ^ (E(a(3t) +E(at*) + a 2 £(f 2 )),
2 «2
Thus
Var (k) =jE{t) +j 2 E(t)* + j 2 E(t>) ^E
= (£(0+j£U 2 )+Var(0)
/ Var(0 y/*
Corr (*,*)=[ t^ )
\E(t) + ±E(t 2 ) + ^V a r(t)/
/ 1 \ 1/2
1  g £(Q  1 E(t 2 )
a Var (t) a Var (*)/
1/2
<
! , /8 g(0
a Var («)/
a
Now — is the expected number of demands per flying hour, which is typically very small, so that strong
correlation will exist only if the variance of t is large relative to its mean. We are thus led to the some
what vacuous conclusion that correlations will be large only if flying hours are extremely variable and
thus cannot be predicted with assurance.
From our earlier discussion of the probability model, it should be apparent that demands are
not statistically independent of flying hours, but it should also be clear that the dependence is dis
tinctly nonlinear. The optimal inventory decisions generated in the model involve highly complex
interactions among the parameters of the distribution, the relevant costs, and flying hours. Predictions
of demand based on simple linear relations between demands and flying hours are overly naive and,
as the discussion here shows, are based on a faulty premise.
IV. AN APPLICATION TO INVENTORY MANAGEMENT
Calculation of Probabilities Required for the Inventory Model
The inventory model employed in this analysis employs dynamic programming techniques to
determine the optimal order size in each period, y<, and the optimal initial stockage, /o, using a single
BAYESIAN DEMAND ESTIMATION 617
state variable, J t , the number of items on hand, on order (but not yet delivered), and in repair at the end
of period t. Defining/r(7) as the total discounted expected costs under control of the inventory manager
from period t to the end of the planning horizon, given J units on hand, on order, and in repair, following
an optimal policy, the following recursive relationship may be used to determine yt and h:
forf=r/! + l, . . .,T+l
where
min {K8(y t )+aEf t + l [Jti+y,D t + R t ]
+ a l >G t + ll (J t  1 + y t )} for*=l, . . .,Th
af l (I )+G'(Io)+K8(h) for* = 0,
Gt + itiJ) = expected holding and penalty costs during period t + l t , given J units on hand, on order,
and in repair at the beginning of period t;
G'(Io) = expected holding and penalty costs during periods 1, 2, . . . , li, plus initial holding
costs, given a starting on hand inventory of /o.
This inventory model is designed to be used with any distribution of demand. Three probability
calculations are required:
1. Probability of k\ failures in n decision periods.
2. Probability of kz nonrepayable failures in m decision periods, given a probability p that a
failed part is repairable.
3. The probability that in two nonoverlapping time intervals, t\ and t 2 of length n and m periods,
respectively, a total of k failures and nonrepairable failures will be observed where all failures
are recorded in t\ and only nonrepairable failures are recorded in h.
The first calculation follows immediately from Equation (6).
If t is the number of flight hours per aircraft per period and the distribution of X, the rate of failures,
is G a , /s, then
(7) P[k failures in one aircraft in one period] = y, „, . \ a  1 e~ 0X d\
Jo k\ I (a)
I ot + kl W W t \ k
\ k )\P+t) \P+t) *
Then if r is the number of aircraft,
(8) P[ki failures in n periods] def =Pn,r(k 1 )
= /nra + k 1 l \ I ft \ nra t t \ kl
\ k, )\P + t) \P + t ) *
This result follows because the Negative Binomial is reproductive with respect to a. Note that we are
modeling the n period, r aircraft process as the sum of n • r independent replications of the basic process
618 G. F. BROWN, JR. AND W. F. ROGERS
in equation (7). An alternative formulation would result if we considered a single process and nrt
flying hours as follows:
D ... f (knrt) k e Xnrt ,_,,..
(9) Pnr(k) = j Q s £j dF(k)
a + k1 \ ( B \ a ( nrt
)\ B+nrt) \
k / \ B + nrt / \ B+nrt
However, the intent here is to incorporate the uncertainty about the value of A. which arises in
large part from the variability and unpredictability of the environments in which individual aircraft
will operate at different times, and thus the representation in eauation (8) is appropriate.
For the second calculation we require the following.
THEOREM. If failures are distributed as the Negative Binomial with parameters a and ( a '_, I
and the probability that a failed part is repairable is p, then the distribution of nonrepayable failures is
again Negative Binomial with parameters a and ( _ ,.. _ — r ]•
PROOF:
00
P[k nonrepairable failures] = ^ P[k nonrepairable failures  x failures]
x=fc
_ / (lp)A * / B \ a 1 ^ (a + x + kl)l . ( pt \*
V B + t ) \B + t) k\^ x!(al)! \B + t)
X=0
(wrwrnsr*: 4  1 )^
= \ k ){fi+(ip)t) (j8+(lp)J Q  E ' D 
,(a+fe)
BAYESIAN DEMAND ESTIMATION 619
Again exploiting the reproductive quality of the Negative Binomial, we have the probability that r
aircraft generate £2 nonrepairable failures in m periods:
nm P (k,)=( mra ~ Hi2 ~ 1 \ ( P Y r " I {1 ~ p)t V 2
(10) PmAk2) \ h )W(ip)t) W(ip)t) ■
The third calculation may now be carried out directly:
(11) P[k l + k 2 = k] = f t P„,r(kj)Pm,rU).
j=o
Calculations are simphfied by use of the following simple recurrence. If K is distributed as the
Negative Binomial with parameters a and b,
/>[£=*+ 1]== ± a b )p[K=k]
P[K=0] = b a .
Empirical Results for F14 Parts
This section contains empirical results from an application of the procedures described in the
preceding section to parts currently being provisioned for the F~14. While a number of the results
summarized here have been predicted by the theory, they give illustrations of the great magnitude
of the effects of these influences.
Table I presents results for the F _ 14 nose landing gear as a function of the degree of uncertainty
about the failure rate.* A wide range of the parameters a and /8 was chosen to illustrate prior distribu
tions all having the same mean, but with increasing uncertainty (or variance). Each of these prior
distributions implies the parameters of the unconditional demand distribution (the Negative Binomial),
which are also tabulated. Finally, three outputs of the inventory model are included:
(1) The optimal initial stockage:
(2) The optimal reorder policy;
which is of the (5, S) form. If A' is the stock on hand, on order, and in repair at the beginning of a period,
the optimal reorder policy is do not order if X 2* s and order S— X if X < s.
(3) The expected inventory system costs,
over a 6month cruise, if an optimal policy is followed (for a deckload of 24 aircraft, each flying an
average of 1 hour per day).
"Similar tables for additional parts appear in INS Study 37.
620
G. F. BROWN, JR. AND W. F. ROGERS
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BAYESIAN DEMAND ESTIMATION
621
The first column in the table corresponds closely with a simple Poisson distribution. The mean
and the variance of the unconditional demand distribution are virtually identical. This results from the
fact that the variance of the prior is extremely small; failure rates different from the prior mean are felt
to be very unlikely and are given little weight. Moving across each table, these results are presented
for cases in which the uncertainty about the true failure rate grows larger; thus the variance in the
unconditional demand distribution also increases. As the uncertainty grows, the inventory system
costs and the required stockage levels also increase rapidly. These empirical results clearly demon
strate the high costs associated with uncertainty about the demand distribution, and show the impor
tance of the management actions which can be taken to reduce this uncertainty. Early in the provision
ing process, it is unlikely that there would be great confidence about the demand rate; thus, if parts
are procured at this time, the high inventory system costs associated with uncertainty must be incurred.
Planning for full deployment of a weapons system before much information about it is gathered could
potentially require support at a cost much higher than would be required later in its service life.
The great costs associated with uncertainty are further illustrated in Table II. There, inventory
system costs are presented for a range of means and variances of the unconditional demand distribution.
A decrease in the mean represents an increase in "reliability," while an increase in the variance
represents a greater degree of uncertainty about the mean. The surprising conclusion that comes from
this table is that uncertainty may be more expensive to the Navy than unreliability. Changes in the
mean (holding variance constant) affect inventory system costs very little, while changes in the variance
(holding the mean constant) produce much greater cost increases. Hence programs to redesign equip
ment may have very little impact unless greater certainty results from the redesign process.
Table II. Effects of Reliability and Uncertainty on Expected Inventory System Costs *
N. Mean
Nv
0.010
0.012
0.014
0.016
0.018
Variancev
0.010
64,716
0.012
71,418
72,003
0.014
77,812
78,305
79,439
0.016
84,010
84,595
85,604
87,142
0.018
89,919
90,755
91,861
93,316
95,196
*For nose landing gear.
Table III shows the effect of changes in the flying hour program on optimal stockage and reorder
policies, and on expected inventory system costs. We have previously shown that, while demands cannot
be predicated by means of a naive relationship with the flying hour program, the flying hour program
does enter in the demand distribution in a complex way, and thus must affect resource allocation deci
sions. These points are clearly demonstrated in the table higher flying hour programs require greater
inventory investment and are much more expensive. Furthermore, the greater the uncertainty about the
system, the greater will be the increase in this investment. Wartime flying hour programs with a system
622
G. F. BROWN, JR. AND W. F. ROGERS
Table III. Effects of Changes in Flight Hour Program*
Flight hours**
Optimal initial
Optimal reorder
Expected inventory
stockage
policy
system costs
0.5
2
(1,2)
57,685
0.75
2
(1,2)
73,000
1.0
2
(1,2)
90,298
1.25
3
(2,3)
105,454
1.5
3
(2,3)
120,837
2.0
4
(3,4)
151,575
2.5
5
(4,5)
180,982
3.0
6
(5,6)
209,899
4.0
7
(7,8)
267,278
5.0
8
(8,9)
321,050
*For nose landing gear, a = 0.056, )8 = 4.0.
**Average daily flight hours per aircraft.
about which there is great uncertainty will require enormous inventory investments. The potential
value of management actions aimed at reducing uncertainty again becomes apparent.
Finally, Table IV illustrates the fact that greater uncertainty about a system also leads to greater
potential wastage. Presented in the table are the probabilities of demands of various sizes on a single
day for a system with mean of 0.014 and the variances listed. As the variance increases, two things
happen: the probability of zero demands increases and the probability of large demands increases.
Thus, while the inventory decision must provide insurance against these high demands and the asso
ciated lessening of readiness, the probability of this insurance being wasted also increases. While
the changes in the probability of zero demands seem small numerically, over an extended period of
time, these small changes become significant. Again, a reduction in uncertainty will lead to a decrease
in both required stockage and potential wastage.
BAYESIAN DEMAND ESTIMATION
623
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REFERENCES
[1] Brown, G. F., Jr. and Perlman, B. L., "Optimal Inventory Management for Naval Aviation Support,"
Center for Naval Analyses, Research Contribution 186 (1971).
[2] Center for Naval Analyses, "F— 14/Phoenix Provisioning: A Study of Initial Spare Part Provisioning
of A New Weapons System," Institute of Naval Studies Study 37 (1972).
[3] DeGroot, M. H., Optimal Statistical Decisions (McGrawHill Book Co., New York, 1970).
[4] Feller, W., An Introduction to Probability Theory and Its Applications (John Wiley and Sons, Inc.,
New York), Vol. I, 1957, Vol. II, 1966.
[5] Raiffa, H. and Schlaifer, R., Applied Statistical Decision Theory (the M.I.T. Press, Cambridge,
Mass., 1961).
READINESS AND THE OPTIMAL REDEPLOYMENT OF RESOURCES*
Seymour Kaplan
Department of Industrial Engineering and Operations Research
New York University
ABSTRACT
This paper considers the problem of the optimal redeployment of a resource among
different geographical locations. Initially, it is assumed that at each location i, i= 1, . . .,n,
the level of availability of the resource is given by a t S» 0. At time t > 0, requirements Rj(t) 3=
are imposed on each location which, in general, will differ from the a t . The resource can be
transported from any one location to any other in magnitudes which will depend on t and the
distance between these locations. It is assumed that 2/?j > 2a;.
The objective function consideis, in addition to transportation costs incurred by re
allocation, the degree to which the resource availabilities after redeployment differ from the
requirements. We shall associate the unavailabilities at the locations with the unreadiness
of the system and discuss the optimal redeployment in terms of the minimization of the follow
n
ing functional forms: ^ kj(Rjyj) + transportation costs, Max [kj(Rj — yj)] + transporta
S=\ i
n
tion costs, and V kj(Rj — yj) 2 + transportation costs. The variables yj represent the final
amount of the resource available at location j. No benefits are assumed to accrue at any
location if » > Rj. A numerical three location example is given and solved for the linear
objective.
PROBLEM
Suppose there are n geographical locations where an organization requires varying levels of a
resource (manpower, fuel, equipment). The requirements for this resource are assumed to change as
sudden demands for the resource brought about by changing economic, political, or natural conditions
are created. For example, natural disasters such as floods may create a need for certain types of rescue
equipment at various flood locations. To satisfy the needs at any one location, the resource may be
obtained locally or from any other locations where availability exists. There are limitations on the
magnitudes of the resource which may be transported from location i to location j. These limitations
depend on the allowable time, t, for reallocation to take place as well as the distance between locations.
In the present problem, t is fixed and given so that the limitations are given constants.
We shall consider several types of objective functions (to be discussed below) which we wish to
associate partially with the degree of unreadiness of the system. That is, we consider several different
measures of unreadiness and investigate how the optimal reallocation changes with these measures.
In addition to the costs incurred as a result of unreadiness, we assume that the physical process of
reallocation also results in transportation costs. The weighted sum of these two types of costs will
♦Prepared under Contract N0001467A04670028 for the Office of Naval Research.
625
626 s  KAPLAN
constitute the objective function. In each case, it is assumed that ending up with more of a resource
than required at a location does not result in any benefits. Also the problem is deterministic and con
tains no stochastic elements.
DEFINITIONS
Let
xij= the amount of the resource to be transported from location i to location j;
yj— the final level of the resource at location,/;
dj= the cost of transporting one unit of the resource from location i to location,/', c y 5 s 0;
a,= the initial availability of the resource at location i, a, 5= 0;
Rj(t) = the requirement of the resource by time t at location./', Rj(t) = Rj 2s 0, where t is assumed
fixed;
Mij(t) = the maximum allowable magnitude of the resource that can be shipped from i loj in an
interval of length t(My(t) =Af« 5 s 0) ;
kj= the relative importance of location j insofar as resource insufficiency at that location is
concerned. The greater kj, the more critical an insufficiency at location j, kj 3 s 0.
I
It is assumed that
PROBLEM FORMULATION
The problem to be solved can be set up in a transportation type format where each location is
considered as both an origin and destination. The constraints state that the amount of product to be
sent from location i cannot exceed a*, the amount received by any location is equal to yj, where yj
cannot exceed Rj and the amount shipped from any location to any other is limited by the My. Thus,
we obtain:
(1) Min z = f(R i ,y j )+^^c ij x i}
i j
n
V x,j =£ a,; i = l ... n
n
2 x v = yr> > =1 ' • • • n
yj^Rj j=l ... n
Xij =£ My all i,j
Xij^O, alii,;; yj ^0j=l, . . . n.
READINESS AND REDEPLOYMENT 627
The objective will be referred to as the unreadiness function and we shall consider and discuss
several different mathematical forms of this function. Note that the yj are problem variables. If we
take a linear objective function of the form
j=i i=i j=\ *
it will be seen that the problem can be reduced to a standard capacitated transportation problem.
Let Uj = Rj — yj. Then (1) becomes:
(2) Min z^kjUj + J, ^ a^j
V xq ^ at i = 1 , . . . n.
j=i
V Xij+Uj — Rj j=l, . . . n
Xij *s Mij
xij 2s 0, Uj^ 0.
If the Uj are considered as the amounts shipped from an additional fictitious origin then the
problem can be considered as one where the unreadiness costs (the kj) are associated with shipping
from the additional origin. If the availability at this origin is considered to be a„ +i , where a„+i may be
set equal to some large value IV Rj will do), then an additional origin constraint of the form
n
^ «i =£ a„+i
i=i
puts the problem into a format with n + 1 origin constraints and n destination constraints. The problem
may be interpreted insofar as unreadiness is concerned, as one where we wish to avoid shipping from
n
origin (n+1) as much as possible. If the V Xij = Rj then the requirement at j can be met without
i=l
n
unreadiness penalty. If V *y < Rj, then a penalty due to unreadiness is incurred at location /. Or,
i=l
one may state the problem as one where unreadiness costs are only associated with slack variables
in the destination constraints when the problem is cast in the form:
n+l n
(3) Min ^ 2 W'
628 s  KAPLAN
subject to
V *y *£ a*, i=l, . . . n+l.
£ Xy^Rj,j=l, . . . n.
i=l
Xij ^ My
XijSzO, i=l, . . . n+l; 7=1, ... ti,
C« +l,j = "J
and where
To finally state the problem in the standard transportation format, consider an additional fictitious
destination such that the slack variables of the origin constraints represent the amounts of the resource
shipped to this destination. Call the slack variables x\, „+i, where i~ 1, . . . n + 1. Then the problem
becomes
n+l n+l
Min 2 £ CijXij,
subject to
2) Xij = at, i=l,2, ... n+l
j=i
n+l
2
i=\
In this problem,
2 x«=/?j, 7=1,2, . . . n+l
*£ xtj =£ Mi } all i, j.
n
fln+l = V ^i
n+l n n / n+l n+l \
fln+l = ^ a « = 2 ^ = 2 °* ( SO tnat ^ a ' = 2 ^ J ) '
i=l j=l i=l \ i=l j=l '
Also the Mjj= Min (a,, /?j) so that if /?j < «j location j will only end up with Rj, whereas if Rj 3 s a/,
the entire availability can remain. Since the xu represent shipping from a location to itself, we shall
assume that ca = 0. Also, we take M n+ i t j = Rj so that if necessary, up to Rj units will be sent to destina
tion 7'from origin n + 1; and M,, n+I = a,. Finally Cj, n+1 = 0, i= 1, . . . n+l.
Assuming that a feasible solution exists, the above problem can be solved as a capacitated trans
portation problem with n + l origins and n+l destinations.
READINESS AND REDEPLOYMENT 629
When the objective is in the form
2= Max [kj(Rjyj)\ + 2 £ ajxij,
we can convert the problem to a linear program, but not a transportation problem by noting that
z=Max\k J (R J y J ) + f t 2 c ijXij ]
After making the transformation u,j = Rj — yj as before, the problem is equivalent to the following
linear program:
(4) Min v
n n
^j"j + X 2 Cy *y ** v ' 7=1, 2, ... n + the other constraints of (2).
i=i j=i
The above objective is often referred to as a minimax objective and can occur in curve fitting and
regression problems as well as in the present context. See [4] for example.
With a quadratic objective of the form
z=2*j(*jr;) 2 +22 w ,
j=\ i=l )=\
the problem may be solved as a quadratic program after letting Uj — Rj — yj, since the quadratic form
is positive definite (kj, cy 2* and the form cannot have the value zero since a, < Rj). Wolfe's method
for quadratic programming is a convenient procedure to use. See [43].
It should be noted that with the minimax objective and the quadratic, the problem can be solved
via simplex tableaus. The minimax problem requires n additional constraints above the n + 1 origin
and n+ 1 destination constraints, where n = the number of locations. The quadratic problem, via the
Wolfe technique requires (n+1) 2 additional constraints, corresponding to the number of variables
in the problem with n + 1 origins and n + 1 destinations.
OBJECTIVE FUNCTION
The objective function is one which transforms the cost of unreadiness into costs associated with
transportation and assumes such a cost is additive to the transportation costs. The great difficulty of
such a procedure is of course in developing meaningful empirical procedures for such a transformation.
630 s KAPLAN
If we consider that the objective functions represent a disutility to the organization then we are assum
ing that the disutility due to unreadiness is additive to that of transportation cost. We are here essen
tially dealing with the problem of decision making with respect to multiple objectives and encounter
the usual difficulties when doing so. See [1] for examples.
In the context of the present problem, we consider the disutility due to unreadiness to be the
major concern and include the transportation costs because the formulation is more general, no diffi
culties are added to the problem in solution, and because such costs may, in fact, influence the optimal
reallocation if some of them are sufficiently large. However, the problem can also be considered with
all Cij= so that the unreadiness disutility is the only consideration.
The linear objective function for unreadiness assumes that the overall unreadiness is measured
as a weighted sum of the insufficiencies in the supply of the resource, the weight taken over the different
geographical locations. The weights may be normalized and could be estimated by a variety of tech
niques relating to the problem of decision making with respect to multiple criteria. In essence, we are
assuming that the organization has an additive linear disutility function with respect to resource
insufficiencies.
With the objective function which minimizes the maximum insufficiency, the measure of unreadi
ness is related to the worst possible insufficiency and is essentially a "conservative" criterion. For
any optimum solution to this problem, the average insufficiency taken over locations will, in general,
be expected to be greater than with the previous criterion.
With the quadratic unreadiness objective, the measure of course penalizes locations more severely
for insufficiencies > 1 than does the linear function. Here again the assumption is of an additive utility
function taken over locations.
Much of which type of objective, of the three discussed, as well as others, will of course depend
on the nature of the resource and how it is combined or used with other resources. Resources such as
aircraft fuel may, in short supply, penalize short run operations much more severely than resources
such as certain food items. In the latter case the minmax objective might be more appropriate since
we might be interested in the shortage of such resources not getting out of "control" anywhere and
trying to keep the worst possible shortage as low as possible.
EXTENSION TO MULTIPLE RESOURCES
If we assume that a simultaneous shortage of two or more resources affects the ability of the
organization to carry out its mission to an extent greater than or equal to that of one resource, then
we can postulate a variety of models for describing this simultaneous shortage.
Much will depend on how the resources interact with each other in carrying out functions. Thus,
certain levels of pilot and airplane shortages simultaneously may not affect the readiness much more
than the given shortage level of just one of these, whereas corresponding shortages of pilots and ASW
equipment may affect the readiness of a unit in an additive manner.
An additive situation would seem appropriate when the resources in question were used for what
may be termed "independent" missions, where the resources needed for one mission are unrelated to
those needed for the others. Of course, in a real sense, no two missions of an organization during a
particular period of time are truly independent. However, if the additive model seems appropriate,
the problem could be handled by including another summation in the objective function over resources
READINESS AND REDEPLOYMENT
631
and adding additional constraints for each resource. Thus, the form of the objective function for the
linear unreadiness model would be:
Min2= i S M*yyy)+i; £ £ <*,*„,,
«=i j=i
/=l i=l j=l
where there are <? resources, and where the subscript / refers to the /th resource.
Nonadditive situations would involve certain nonlinearities in formulation and are beyond the
scope of this paper.
EXAMPLE
We shall illustrate the solution for the linear objective function with an example. Consider the
following reallocation problem with three locations, set up in a tableau format as follows:
Location
1
2
3
a,
1
4
0.01
2
0.02
2
4
2
0.02
3
6
0.02
3
6
3
0.02
1
0.01
1
7
7
Rj
6
8
8
2aj=17
ki
0.4
0.3
0.2
2Rj = 22
The numbers in the upper left of each cell of the 3X3 location matrix indicate the transportation
costs, while those in the lower right indicate the capacity of each route, i.e., c 1 2 = 0.01, %\i =£ 2. The
overall requirement is for 22 units, whereas the overall availability is 17.
We shall solve the problem by means of the primaldual method for the capacitated transportation
problem and the notation and tableau format of Hadley [2].
The problem requires six tableaus for solution; they are shown in the appendix. The optimal
minimum cost solution is found by transporting one unit from location three to location one and one
unit from location three to location two. The optimal redeployment can be read off the final tableau
(Table 1) reproduced below. The values in the circles of the fourth row cells (0 4 ) corresponding to the
fictitious origin, show the final deficiencies at each location, i.e., R\— yi = l, /?2 — J2= 1, /?3 — J3 = 3.
(The 17 is the excess going to the fictitious destination). The values in the circles on the offdiagonal
elements indicate the redeployments. In this problem the value of the objective function is z mln = 1.33.
632
S. KAPLAN
Table 1. Final Tableau for the Example.
For notation, see pp. 358 and 397 of [2].
Solution *3i = 1, x 3 2= 1, z— 1.33.
Os
0,
M
D
2
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3
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4
0.00
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7
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6
0.30
8
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8
0.00
22
22
©
©
0.10
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(2)
bj
6
8
8
17
«j
Pi
READINESS AND REDEPLOYMENT
633
D,
APPENDIX
Tableau 1
D 2 D 3 D t
o,
o 4
VJ
n n
a c\
n n
A A
a (
8,
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u, \
*;
0.0
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0.00
7
7
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22
5
4
©
5
Rj
6
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8
17
5
X
0}
2
2
1
«J
5
Pi
4
/i = 0.20
634
S. KAPLAN
£>■
Tableau 2
D 2 D 3
D<
0,
o 4
vj
0.0
0.20
Of
8<
■y<
Ui ^V
U V
*\
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0.00
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22
4
4
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4
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6
2
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17
4
*>j
2
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4
4
Pi
4
4
A = 0.01
READINESS AND REDEPLOYMENT
635
Tableau 3
o,
0,
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02
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x>.
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A = 0.01
636
S. KAPLAN
0.
Tableau 4
z> 2 d 3
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o,
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READINESS AND REDEPLOYMENT
637
Tableau 5
o,
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0.10
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h = 0.l0
638
S. KAPLAN
0,
0,
Q.
Di
Tableau 6
A D 3
D 4
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0.00
0.20
0.40
at
x (
8,
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0.00
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6
8
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Pi
REFERENCES
[1] Fishburn, Peter, Decision and Value Theory (John Wiley and Sons, Inc., 1964).
[2] Hadley, G., Linear Programming (AddisonWesley Publishing Company, Inc., 1962).
[3] Hadley, G., Nonlinear and Dynamic Programming (AddisonWesley Publishing Company, Inc.,
1964).
[4] Wagner, Harvey M., Principles of Operations Research (PrenticeHall, Inc., 1969).
ON MAXMIN PROBLEMS
Kailash C. Kapur
Department of Industrial Engineering and Operations Research
Wayne State University
Detroit, Michigan
ABSTRACT
Necessary and sufficient conditions for maxmin problems are given here. In addition to
characterization of directional derivatives of the relevant functions, subdifferentiability set
for such functions is characterized.
INTRODUCTION
Let/(x, y) be a real valued function of x and y, where xeX C E" and y belongs to some compact
set Y of a topological space. Let us assume that f(x, y) and its partial derivatives with respect to x
are continuous in x and y taken together. Let:
(1) <j>(x) = Minf(x,y).
ytY
Then, the maxmin problem is
(2) Max <M*) = Max Min/(*,y).
xtX xtX ytY
Maxmin problems arise very often in system sciences and cybernetics, operations research,
economics, game theory, optimal control theory, and duality in mathematical programming. If Y is a
finite index set {1, 2, . . . p}, then <\>{x) is the scalar function whose value for each x is the least
among the p values of/, which may be denoted by fi (x) , f 2 (x) , . . ., f P (x). Hence, problem (2) is
to find the maximum of this minimum. This is known as a minimum component maximum problem [5].
Similarly, in control theory and design of systems, we have to consider the problem of minimizing the
maximum error or deviations from the desired values of the parameters of the system [7]. For applica
tions in economics, game theory and allocation problems in defense analysis, see Danskin [3, 4] and
for maxmin problems which arise in duality theory in mathematical programming and control theory
see Lasdon [6] and Luenberger [7, 8].
The objective of this paper is to give some necessary and sufficient conditions for optimality for
maxmin problems which are generalizations of the results due to Danskin [3, 4].
OPTIMALITY CONDITIONS
It is well known that the function </> in (2) is not differentiable, but has directional derivatives. A
necessary condition for local optimality at a point x° is that the directional derivatives be nonpositive
in every admissible direction. An element yeE" will be called an admissible direction at a point x°eX if
639
g40 K c  kapur
there exists a sequence {y n }, y"eE",and a scalar sequence {a„}, such that
a) (x°+a n y»)€X
b) y"»y
c) a„>0 and a« »0.
The set of all admissible directions is a closed cone and denoted by C(X, x°). For a detailed discus
sion, see [1, 4. 10].
Let
(3) Y(x°) = {y€Y\f(x°,y)=<Hx )}.
Then, the directional derivation of 4>(x) at JC°eA'in the direction y,y = 1, is given [3] by
(4) D y <t>(x°)= Min
ytY(x°)
A a/(*,y) j =Min [y , V /(x, y )].
Dy<j)(x) is continuous in x and y and convex in y.
LEMMA 1 : If x° is a solution for (2), then
(5) Dy<t>(x°) ^0 for all yeC(*, x°).
PROOF: For^°+a n y"e^,
<l>(x + any") = <f>(x°) + Dyn<}>(x o )a„ + 0(a„)
and hence Dyn<(>(x o )an + 0{an) ^ 0.
When dividing by a„ and taking the limit, the proof follows. Q.E.D.
By using (4), (5) can be written as
MaxMin [y, V/(*, y)]^0
y=J ytY(x°)
or
MaxMin [y, Vf(x, Y)] =0.
ys0 ytY(x°)
The sufficient condition, which is easy to prove, is given by Lemma 2.
LEMMA 2: If is concave and X is convex, then (5) is also sufficient for x° to be a solution of
MAXMIN PROBLEMS 641
(2). (5) in this case can also be written as
MaxMin [(xx°),f(x, y)] = 0.
XtX «€V(X°)
Though function <f> is not differentiable, it does have subgradients, and we develop an expression
for them here.
A vector yeE n is said to be subgradient of a concave function <f> at a point x° if
<f>(x)<f>(x°)^[(xx°),y] for all*.
The set of all subgradients of <f> at a point x° is called the subdifferential of <f> and denoted by
d<f>{x°) , which is a closed convex set. In [9], it is proven that
(6) Dy<t>(x) = Inf [yy\yed4>(x) ] .
Let W(x°) denote the set of vectors V/j>°, y) with yeY(x°) and let W(x°) be the convex hull of
W{x°). Then, W (x°) is a compact convex set.
THEOREM 1: d<f>(x°) = W (x°).
PROOF: When y€Y(x°),<f>(x°)=f(x°,y) and
Hence <M*) =/(*, y) for all x.
^{x)<f>{x°)^f{x,y)f{x°,y).
^[(xx°),W(x°,y)]
because /is concave in x. Hence, V/(ac°, y) is a subgradient of <f> and W(x°)Cd<f>(x°). Hence,
W(x°)Cd<f>(x°) because d<f>(x°) is a closed convex set. To prove W(x°) Dd<l>(x ), let yed<f>(x°) and
ytW{x°). Hence, by separating hyperplane theorem [7], there exists a hyperplane y* strictly separating
y and W(x°), because W(x°) is compact convex set.
Thus
y* . y < y* . y f or all y<W (x°)
and using (6),
D y *<f>(x°) ^ y* ■ y < Min y*y = D y *<f>(x°),
ytw(x°)
which is impossible. Hence, W(x°) D d<M*°) and W (x°) = d<f>(x°). Q.E.D.
542 K c KAPUR
Thus, by the above theorem, yeE" is a subgradient of <j) at x° if, and only if,
n+l
(7) y=Y^f(x°, yi ),
where
V Xi= 1, Ai s* and yieF(*°) for all i.
i=l
The above follows from Caratheodory's Theorem [9] which states that a point in the convex hull
of a set in E n can be represented by a convex combination of at the most (n+l) points of the set.
For a general optimization problem, let the set X(lE n be given by
(8) X={x\gi(x)^0, i'=l, . . ., m andr;(x°)=0, i=l, . . . k} ,
where gi and n are differentiable functions.
Let L be the subspace spanned by Vn(.x ), i= 1, . . ., A: and L^ be the orthogonal complement
of Lq. Also, let
Cx°= {*l [*, V#(*°)] ^ 0, for all i such that gt(x°) = 0}.
For the necessary conditions, let at a point x°, the following constraint qualification be satisfied:
(9) (C x °nLj i ) = C(X,x ).
For details about generalized constraint qualification see [2, 10].
THEOREM 2: Let at a point x°, the constraint qualification given by (9) is satisfied. Then if x° is
n + l
an optimal solution of problem (2) there is some nonnegative vector X, Xi = 0, V Aj= 1, such that
n+l m fc
J \,V/(*°, yi ) = £ iw,(*°) +5>r,(*°)
t=l i=l i=\
Uigi(x°)=0, 1=1, . . ., rn
Ui ^ 0, vi unrestricted and yieY(x°).
If/ is concave function for all yeF and the set X is convex, then the above conditions are also suf
ficient for x° to be a solution of problem (2).
PROOF: By slight generalization of the method given in [1] , it can be proved that (using Theorem
1 and Equation (9)):
(C xa nL^)*nd<f>(x°)+(}>,
MAXMIN PROBLEMS 543
where * denotes the polar or dual cone [7] of the cone (CxoHL^). Hence, for •ye(C jr .ni4)*na$(*°),
we have
y=5Sv/(x°,y,)
t=l
and ye(C x ° D L±)* which implies that,
[y , S/gt (x° ) ] ^ for all i such that gi (x°) =
[y, Vr,(* o )]^0 forall;=l, . . ., *
[y,Vn(*°)]^0 for aU i=l, . . ., k.
Hence, by Farka's Lemma, we have
£ \iVf(x\yt)=2 UiVgiix )^ (v tl v a )Vri(x ),
i=l i i=l
where «i ^ and va, va ^ 0; and let u, = when gK* ) < 0, we have
n+l m k
£ ki\7f(x°, yi )=V u i V gi (x°)+2 ««Vr,U°)
i=l i=l i=l
u igi (x°)=0
m^O and Vi unrestricted.
Thus, the above theorem shows how maxmin problems can be reduced to a problem of single
function optimization problem where at the most (n+l) functions are to be considered and they are
added together by weights A.,. Hence, necessary and sufficient conditions for problem (2) are equivalent
to the conditions for
n+l
(10) Max V K,f(x,y t ),
x*x i=1
n+l
where y,eF(*), £ Xi=l and x > =°
i=l
CONCLUSIONS
The optimality conditions given above can also be generalized for maxmin problems in infinite
dimensional spaces. The characterization of the subgradient can aid the development of computational
methods for such problems.
644
K. C. KAPUR
REFERENCES
[1] Bram, J., "The Lagrange Multiplier Theorem for MaxMin with Several Constraints," J. SIAM
Applied Mathematics, Vol. 14, No. 4 (1966), pp. 665667.
[2] Canon, M., C. Cullum, and E. Polak, "Constrained Minimization Problems in Finite Dimensional
Spaces," SIAM Journal Control^ 367389 (1969).
[3] Danskin, J. M., "The Theory of MaxMin with Applications," J. SIAM Applied Mathematics,
Vol. 14, No. 4 (1966), pp. 641664.
[4] Danskin, J. M.,The Theory of MaxMin (SpringerVerlag, New York, Inc., 1967).
[5] Kuhn, H. W., and A. W. Tucker, "Nonlinear Programming," Proceedings of Second Berkeley
Symposium (University of California Press, Berkeley, 1951), pp. 481492.
[6] Lasdon, L. S., "Duality and Decomposition in Mathematical Programming," IEEE Transactions
on Systems Science and Cybernetics, Vol. SSC4, No. 2 (1968), pp. 86100.
[7] Luenberger, D. G., Optimization by Vector Space Methods (John Wiley and Sons, New York, 1969).
[8] Luenberger, D. G., "Convex Programming and Duality in Normed Space," IEEE Transactions on
Systems Science and Cybernetics, Vol. SSC4, No. 2 (1968), pp. 182188.
[9] Rockafellar, R. T., Convex Analysis (Princeton University Press, Princeton, N.J., 1970).
[10] Varaiya, P., "Nonlinear Programming in Banach Space," SIAM Journal of Applied Mathematics
15, 284293 (1967).
A THEORY OF IDEAL LINEAR WEIGHTS FOR HETEROGENEOUS
COMBAT FORCES*
David R. Howes
U.S. Army Concepts Analysis Agency
and
Robert M. Thrall
Rice University
ABSTRACT
Detailed combat simulations can produce effectiveness tables which measure the
effectiveness of each weapon class on one side of an engagement, battle, or campaign to
each weapon class on the other. Effectiveness tables may also be constructed in other ways.
This paper assumes that effectiveness tables are given and shows how to construct
from them a system of weapon weights each of which is a weighted average of the effects of
a given weapon against each of the enemy's weapons. These weights utilize the Perron
Frobenius theory of eigenvectors of nonnegative matrices. Methods of calculation are
discussed and some interpretations are given for both the irreducible and reducible cases.
INTRODUCTION
In conducting military operations research, analysts frequently make use of indices of force
effectiveness which are intended to measure the contribution of some force component to the overall
power of a military force in some hypothetical military conflict. An example of such an index is the
"Firepower Potential" which has been used in a number of U.S. Army analyses as a measure of force
strength. t In the alternative considered here, indices are derived from interweapon effective matrices
(tables) such as might emerge from a detailed combat simulation or from other sources (see for ex
ample, [1], [2], [3], [4], [15], [21], [22]).
When such tables are given it is possible to construct from them a system of weapon weights each
of which is a weighted average of the effects of a given weapon against each of the enemy's weapons.
This paper will describe the construction of such weights.
1. Effectiveness Matrices
Weapon effectiveness may be considered a function of casualtyproduction which lies in depriving
the enemy of the value of weapons lost (cf. [13]). Therefore, it is appropriate to consider numbers which
measure the killing power of each weapon against each opposing weapon. An effectiveness matrix may
be regarded as a table whose entries are these killing powers or relative effectivenesses.
* The original version of this paper appears as Part B of chapter 2 of [26] and was presented at the 1 1th U.S. Army Operations
Research Symposium, May 1972. The authors wish to thank the referee for many helpful suggestions.
tSeveral references on current procedures are cited below; some others (e.g., [Ill, [16], [28]) are included among the ref
erences without having been cited in the text. A full exposition of past efforts at constructing indices of effectiveness would
require access to many classified or otherwise unavailable sources and would go far beyond the scope and puropse of the present
paper.
645
646
D. R. HOWES AND R. M. THRALL
More precisely, consider a combat situation between two opponents, Blue and Red. We suppose
that Blue has m classes of weapons and consider the Blue force vector
(11)
U B =
Uib
_ U>mB_
where U\b is the number of Blue weapons 01 class 1, . . ., UmB is the number of Blue weapons of Blue
class m. Similarly, suppose that Red has n classes of weapons and that
(1.2)
LU„rJ
is the Red force vector.
In the discussion which follows, it is assumed that the Blue and Red vectors of weights are to be
derived in some way from certain interweapon effectiveness matrices; however, there are precedents
constructing weight vectors based directly on other considerations. For example, various military opera
tions research organizations (i.e., CORG, RAC, STAG) have from time to time constructed weight
vectors based on a consensus of military judgement, individuals being asked to score lists of weapons
of types of military unit. Other systems of weights have been based on such considerations as World
War II casualties to personnel materiel or on the average damage radii observed during proving ground
tests of ammunition (see, for example, [6] and [20]).
The effectiveness matrix concept is connected with Lanchestertype theory of combat in section 5.
We wish to find Blue and Red weight vectors
(1.3)
such that the linear combinations
W B =
W\B
W mB
W R =
WlR
W„R
(1.4)
and
S(B) = WibUib+ ■ ■ ■ +WmBUmB=WWB
(1.5)
S(R)=WirU 1R + . . . +WnRUnR=WHJ R
are good measures of the respective overall strengths of Blue and Red. Then the fraction
(1.6)
T=S(B)IS(R)
IDEAL LINEAR WEIGHTS
647
can be used as an index of the relative strengths.
A BluevsRed effectiveness matrix M BR is a matrix (table) having m rows and n columns where the
element m B R(i, j) measures the effectiveness (killing power) of a single weapon of Blue class i against
Red weapon class j. Similarly a RedvsBlue effectiveness matrix
(1.7)
M RB = [m RB (j,i)]
has n rows and m columns and, inversely, m.R B (J, i) measures the effectiveness of a single Red weapon
of classy against Blue weapon class i. The numbers m B R(i,j) and m RB (j, i) may be positive or zero, but,
by definition, cannot be negative.
For example, suppose that m — n = 2, that both Red and Blue weapon class one are infantry weapons
and that both Red and Blue weapon class two are artillery weapons. Then the effectiveness matrices
(1.8)
0.5
0.6
BR
, «L
0.7
0.2
0.6
0.1
would describe a situation in which (1) in infantry combat Red was more effective than Blue (0.6 vs 0.5),
(2) neither infantry could harm the enemy artillery, and (3) the Blue artillery is superior to the Red artil
lery, and (4) each artillery battery has a positive effectiveness against its counterpart.
The effectiveness matrices
(1.9)
M\ R =
0.5 0.1
0.7 0.2
M%b
0.6 0.2
0.6 0.1
would describe a change which gave each infantry capability against the opposing artillery.
The matrices
(1.10)
m%r
0.5
0.7 0.8
MIb =
0.6
0.6 0.5
would describe a different type of change in which the artillery attritions are substantially increased.
If we assume that the artillery units are either concealed or out of each other's range then we could
have effectiveness matrices
(l.H)
MJ* =
0.5
0.7
"0.6 0"
M% B =
0.6
2. Ideal Linear Weights
We turn next to consideration of suitable weight vectors, W B and W R . These should be derived in
648
D. R. HOWES AND R. M. THRALL
some reasonable way from the corresponding effectiveness matrices, M BR and M RB .
For example, one could simply let W B be the average of the columns of M B r. Using M BR and M l RB this
would give
(2.1)
" B
0.5 +
0.7+0.2
0.25
0.45
and W\ =
0.3
0.35
similarly from M BR and M RB we would obtain
(2.2)
"0.3 "
"0.4
w\=
' w R
0.45 j
0.35
This naive approach has the advantage of simplicity, but lacks credibility since it places equal emphasis
on effectiveness against enemy infantry and artillery whereas one of these might be considered much
more dangerous than the other.
The naive approach places equal weight on each column. A more general procedure is to select as
weights nonnegative numbers which add to one. Thus in example 2, if we consider enemy artillery to be
twice as important a target as enemy infantry, we would choose weights 1/3, 2/3 and get
W\ =
1/3(0.5)+ 2/3(0.1)
1/3(0.7)+ 2/3(0.2)
= 1/3
0.7
1.1
A vector with nonnegative elements that sum to one is called a probability vector. Then the more
general procedure would consist of selecting two probability vectors
(2.3)
Z B =
Z B \
z Bm
Zr =
Zr\
ZRn
and then defining the linear weights by
(2.4)
W b = M br Zr, W r = M rb Z b .
We observe (i) that (2.4) gives each weighting factor wis as a weighted average (probability combina
tion) of the effectiveness numbers corresponding to the ith Blue weapon type, and (ii) that the same
weighted average is used for all i. A still more general procedure would be to permit a different
weighted average for each i; this would replace (2.4) by
(2.5)
WiB=^Tn BR (i, j)z R (i,j), WjR=^m RB (j, i)z B (j, i).
IDEAL LINEAR WEIGHTS
649
where all columns of the matrices Zr and Z B are probability vectors.
Returning now to (2.4) the next step is selection of Z B and Zr. In the naive approach we took
(2.6)
Zr— E m — ~
m m
Zr— En—~
n n
Here (and later) we use the symbol E p to represent the column vector consisting of p ones, e.g.,
£3 =
A second, somewhat more reasonable selection is
(2.7)
where
Z B = M B RE n lyR, ZR = M RB E m lyR,
then
y B = E T n M RB E m = ^m„ B (j, i), yR = El l M BR E„=^ m BR (i,j)
>,j
i,j
(2.8)
W B = M BR M RB E m \yR , W R = M RB M BR Enly B .
In Example 2 this gives
(2.9)
72 =
0.6
0.9
1.5, Z\ =
0.8
0.7
1.5
and
(2.10)
" B '"BR^R ''
0.47
0.70
1.5, W\
0.54
0.45
1.5.
In (2.7) the /th component of the averaging vector Zr is proportional to the sum of all Red effective
ness numbers corresponding to the j'th Red type. This tacitly assumes equal importance for all Blue
weapon types. Clearly, we could modify (2.7) by selecting any nonnegative linear combination Mr B V b of
the columns of M RB and then taking Zr as the unique probability vector proportional to M RB V B . The ideal
650 D R HOWES AND R. M. THRALL
linear weights which we next introduce correspond to the choice V B — W B , V r = W r .
To motivate this choice we consider the following argument. Suppose that W R has been determined;
this means that relative values for the Red weapon systems are known. Then it seems reasonable to
select as Z R the unique probability vector proportional to W R . Similar reasoning would apply in selec
tion of Zb if W B is given. This line of argument would lead to
(2.11) Z B =r B /a B , Z R =W R \a R
where
a B = ElW B , a R = ET„W R .
then, we get
(2.12) W b = M B rZr = M br W r Ioir, W r = M rb Zb= MrbW b I<xb,
and by substituting each of these equations in the other we get
(2.13) W b = M br MrbWbIol b <x r , Wr=M rb M B rWrIolr<x b .
Now, let
(2.14) P B = M BR M RB , Pr=M br M B r, \ = a B a R
and we have the equations
(2.15) PbW b = kW B , P R W R = KW R .
The ideal weights must satisfy these equations and also be nonnegative vectors (and also nonzero).
At first glance it might seem that (2.11) and (2.12) involve a circular logic since each of the weights is
ultimately (cf. 2.15) defined in terms of itself. However, this is a familiar situation in mathematics
and is a characteristic of eigenvalue problems which crop up in a wide variety of mathematical models.
In particular, Equations (2.15) are well known in linear algebra. First, they require that X be an eigen
value of each of the square matrices P B (mXm) and \P R (nXn) and that W B , W R be eigenvectors.
Since the effectiveness matrices M BR , M RB have nonnegative elements, the same is true of their products
Pb, Pr.
The classical PerronFrobenius theory of eigenvalues and eigenvectors of nonnegative matrices
applies to our situation and guarantees solutions to (2.14) with W B , W R nonnegative and X 'positive.
Moreover, it follows from the general theory of matrices that P B and Pr have the same nonzero eigen
values. The pertinent facts from the classical PerronFrobenius theory can be found (with proofs) in
chapter XIII of Gantmacher, Vol. II [10]. This chapter also has a comprehensive bibliography (see
also Varga [29]). The original papers by Perron and Frobenius appear, respectively, as References
[19] and [8], (see also [9], pp 404414 and 546567, [14], [27]).
IDEAL LINEAR WEIGHTS 651
3. Examples of Ideal Weights
We return to our four examples to illustrate the theory.
EXAMPLE 1.
(3D p. = r° 30 ° i /> 1= r° 30 ° 1
1 ' B L°54 0.02 J' " [0.37 0.02 J
The eigenvalues for both P\ and P\ are X\=0.30, X» 2 =0.02. Then
(3.2) 7i = r° 34 1 7i _r°43i
£ » L0.66J' Z « Lo.57 J
are the unique probability eigenvectors corresponding to X/ . The corresponding weights are
(33) ri=M'jp = r o  215 l ri = r° 204 i
{A  A) w * m **Fr _ 0.415 J' W * 1.0.270 J
a\ = 0.63 , a* = 0.474, a* a^ = \' = 0.3.
The second eigenvalue \\ gives
0.4) 2i'=*i*=[J]. ^*=[o. 2 ]. *".'[£,]■
We will see later that this second eigenvalue yields less meaningful weights than the first.
EXAMPLE 2.
(3 5) p 2 = [0.36 0.11] p2 = [0.44 O.iol
Ka  0) r B I 0.54 0.16J' R I 0.37 0.08J
The characteristic equation for both matrices is
(3.6) X 2  0.52X  0.0018 = 0,
and has as its roots the eigenvalues
(3.7) X 2 = 0.5235, X 2   0.0035.
From X 2 we get the unique probability eigenvectors
rO.401 „ rO.5451
652
D. R. HOWES AND R. M. THRALL
for P 2 B and P 2 R respectively.
(3.9)
EXAMPLE 3.
(3.10)
W 2
B
ro.321
" lo.48.r
W \
0.36
0.30.
a:
0.8,
a 2 = 0.66, a 2 a 2 = 0.528 ~ k 2
H an
"0.3
P 3 =
B
.0.9
o I p3 = r  30 ° l
0.4 J' R L0.65 0.40 J'
A 3 = 0.3,
i
A 3 = 0.4.
2
This example differs from Example 1 since this time the second eigenvalue is larger than the first
and hence A 3 does not correspond to probability eigenvectors (cf. [10] vol. 2, p. 66). The only prob
ability eigenvectors come from A 3 and are
(3.11)
These give
*.ra .[?]
(3.12) ^ = [o.s]' r " = [o.5]' "£ =0  8 < « 3 «=0.5, « 3 <4=0.4 =
A 3 .
EXAMPLE 4.
(3.13)
ro.30 0]
B [ 0.42 Oj '
r R
0.30
0.30
a
A 4 = 0.3, A 4 2 =0.
This example resembles Example 1 in that the first eigenvalue is larger than the second.
From the first eigenvalue we get
^0^ = 0.3 = A 4 .
The second eigenvalue gives
(3.15)
and thus does not provide useful weighting vectors.
IDEAL LINEAR WEIGHTS
653
Example 2 illustrates a general class of situations where each Blue weapon system is (at least
minimally) effective against each Red one and vice versa. If a square matrix P has positive (not merely
nonnegative) elements then it has a unique probability eigenvector Z and the corresponding eigen
value A, (called the Perron eigenvalue) is not only positive, but has the largest absolute value of all
the eigenvalues of P. It is then easy to calculate Z and \ x by the following sequential process (see [14],
pp. 151152 or [27], p. 250). Let V = E m (where P is m X m), let a (V )=ElV = m, let Z = Vola(V ),
and proceeding inductively let V i+l = PZ u let Z i+ i = Vi +1 la(V i+1 ), i= 1,2 . . ..Then
(3.16) Z=limZ„ \ 1 = lima(r i+1 ).
J— + 00 j— > 00
These results still hold (see [10], vol. 2, p. 80) even iff has some, but not too many, zero elements (i.e.,
as long as P remains irreducible and primitive).
Indeed, when P B and Pr are positive, we can use a limiting process to define the ideal weights
W B , W R .
We can begin with W R any positive vector (e.g., W R = E n ) then in turn setZ£ = fF°/a(Jr°), W\ =
MbrZ*, Z° = WlJa(Wl), and proceeding inductively
(317) W r = Mr B Z b ~\ Z.^W R la{W R )
W b = M br Zr, Z B = W B la(W B ), i=l,2
Then the six sequences
(318) W R , Z R , W B , Z' B , a(W R ), a{W B )
converge, respectively, to
(3.19) W R , Zr, W B , Z b , a«, a B
where Z b ,Zr are the unique Perron probability eigenvectors of P B , Pr, respectively; Wr, W b are the
ideal weights for R, B, respectively; W R = olrZ r, W B = a B Z B , and Ai = a R a B is the Perron eigenvalue
for both P B and P R .
This approach provides a computationally convenient algorithm for calculating the ideal weights.
When m and n exceed two, this approach is clearly preferable to calculating and solving the char
acteristic equation for P B or Pr. There are other more refined computational algorithms which are, in
general, more efficient than this one. However, a computer program written for this iterative process
gave quite satisfactory numerical results for moderate values of m and n. An example involving 40
weapon types converged in nine iterations to an accuracy of 0.0001.
4. Interpretation of Reducibility
Examples 1, 3, 4 illustrate some of the possible effects of zeros in P B , Pr. All of the P's in these
examples are what is called reducible. A nonnegative square matrix P is said to be reducible if it has
the form
654 D  R HOWES AND R. M. THRALL
'[£ y.
where Pi and P2 are square, or more generally, if this form can be obtained by a reordering of the
rows followed by the same reordering of the columns.
In our combat context, we encounter reducible matrices when as in Examples 1, 3, 4 there are
two classes of weapons on each side and the first class of Blue is totally ineffective against the second
class of Red and vice versa.
Let us assume that both Pg and Pr are reducible with Pb\, Pb2, Pri, Pr2 all positive, that Pb\,
Pri have the Perron eigenvalue \i, and that Pb2, Pr2 have the Perron eigenvalue X2. [These assump
tions all hold for Examples 1 and 3.] Then, if we apply our computational algorithm beginning with
W R = E n , the limiting eigenvectors obtained will correspond to the larger eigenvalue.
Thus, in Example 1 we would get W^, W R and not W B *, W R *. In Example 3 we would, of course,
get W\ , W\ and in this case there is no possibility of positive ideal weights.
Moreover, in Example 1 the only way to get the starred vectors would be to start with W R of the form
, i.e., almost all starting vectors W° R will yield W l B , W R . For this reason we choose to limit the
term "ideal" to W\ , W R .
There is a possible interpretation for the different types of weights found in Examples 1 and 3.
In Example 1 the attrition of infantry is so much greater than that of artillery that we visualize one phase
of the battle ending when one side has lost all of its infantry even though both sides still have artillery
left. However, at that time the starred weights do become relevant for the ensuing artillery duel.
On the other hand, in Example 3 the artillery attrition is more rapid than that of infantry. Moreover,
when one side runs out of artillery the remaining infantry forces will ultimately be anihilated by the
surviving artillery. Hence a zero weight for infantry is not inappropriate.
Example 4 is much like Example 1 for even though Pr2 = P«2 = the larger eigenvalue k* still
gives a viable ideal weight.
5. Calculation of Effectiveness Matrices and an Application to Lanchester Theory
There are several possible approaches to calculation of the effectiveness matrices. Only one of
these will be discussed in the present paper.
A sufficiently detailed combat simulation can be expected to produce loss matrices
( 5 l) L B R=[l B R(i,j)], LBR=Ul)R(j,i)],
where Ibr(i,J) is the number of Red weapons of class j lost by action of Blue weapons of class i, etc.
Then we may define effectiveness matrices M B r, Mr B whose elements are the effectiveness numbers;
(5.2) mBR(i,j) = lBR(i,j)lu iB , m HB (j, i) = Irb{J, i)lu jR
where U B and Ur are as in section 1 (formulas (1.1) and 1.2)).
The 1Mb and Ujr might refer either to the initial Blue and Red strengths, or to certain average
strengths during the battle. The choice of an appropriate average would relate to questions not con
IDEAL LINEAR WEIGHTS 655
sidered here; however, a simple case of such an average might be [u«(t=o)+ttj«(f=( I )]/2 where t x is an
arbitrary time chosen as a unit of measurement. The interval (0, ti ) must, of course, not exceed the battle
length and should be small enough so that combat losses have not yet changed the character of the
encounter.
This procedure has as its main drawbacks (1) that the validity of the results obtained depends on
the simulation scenario, on the simulation model, and on the extent of sampling error, (2) that it fails to
consider military appurtenances which, although affecting the combat action, do not cause attributable
casualties to opposing weapon systems, and (3) that it does not take into account scale factors (i.e., it
tacitly assumes that the losses are strictly proportional to the number of weapons in a class).
Effectiveness matrices calculated as above are closely related to the Lanchester parameters
appropriate to a heterogenous Lanchester linear system and could be interpreted as estimates of such
parameters. Such a system represents an extension of the formulae which F. W. Lanchester [17] used
to describe the attrition inflicted on each other by two hostile forces to the case where each force is
composed of various subelements. In such cases, each force can be represented as a vector of elements
and the (scalar) Lanchester attrition coefficients have as counterparts matrices whose elements describe
the interacting effects between the elements. These attrition matrices, if known, could serve as exam
ples of effectiveness matrices as discussed in this paper. Conversely, effectiveness matrices, when
based on data from real or simulated combat, might be interpreted as Lanchester parameters as noted
above.
The generalization of Lanchester equations to the heterogenous case was explored by Snow [23],
then by Dolansky [7], and by Bonder and Farrell [5]. It should be noted that effectiveness matrices may
be derived in other ways and also that the statistical problem of parameter estimation from sample
data is far more complex than might be suggested by the discussion given here.
Dare and James, in Defense Operational Analysis Establishment Memorandum M7120 have made
an analysis based on a Lanchester interpretation with results parallel to those which follow next. In
Tab E, Appendix II to Annex L of the TATAWS III study, BAARINC Inc. has based a similar analysis
on another interpretation.
More specifically, if we have the Lanchester systems
(5.3) Ub = C r Ur, U r = C b Ub
then the (i,j) element Cr(i,j) of Cr represents the effectiveness of R weapon ;' against B weapon i, i.e.,
CR{i,j) = mR B (j, i).
Reasoning similarly for Cb we conclude that
(5.4) Cr = M' b , C B = Ml R
are reasonable choices for the Lanchester coefficient matrices.
Now, differentiating Equation (1.4) with respect to time we get
g56 D R HOWES AND R. M. THRALL
(5.5) S(B) = WlU B = jriM* RB U R
={M RB w B yu R
= {M RB M BR Wnl<x R ) T U R
= —  WIU R = a B W1U R (since A = a R a B ).
a R R R
Now substituting from (1.5) this gives
(5.6) S{B)=a B S(R).
Similarly, differentiating (1.5) yields
(5.7) S(R)=a R S(B).
Equations (5.6) and (5.7) are the ones obtained by Dare and James. A note of caution is appropriate
here. The heterogenous systems (5.3), (5.4) are not valid past the time t* at which any component of
U R orU B becomes zero. Although the summarizing homogenous systems (5.6) and (5.7) will in general
yield solutions S(B), S(R) which both remain positive far beyond t*, the attritionrate coefficients a B
and a R must be modified whenever the weights W B and W R change due to the annihilation of a target
type (see the discussion of Example 1 in section 4).
6. A Larger Example
An example of extended calculation is given below based on results obtained in a particular
detailed war game. No claims are warranted concerning the representativeness of these results,
which are dependent on the particular scenario, and the random statistical variation inherent in the
game model used. Weapons classes for both sides were the same. They were (following some aggrega
tion of similar type):
1. Small arms
2. Armored personnel carriers
3. Tanks
4. Armed reconaissance vehicles
5. Antitank weapons
6. Mortars
7. Artillery
Red forces were in the attack, Blue in the defense.
7 Red Weapons 7 Blue Weapons
IDEAL LINEAR WEIGHTS
657
Red Effects
(6.1)
M RB =
Blue Effects
(6.2)
M BR =
(6.3)
Pr =
(6.4)
Pb
0.0145
0.0012
0.0000
0.0229
0.0004
0.0000
0.0000
0.0510
0.0326
0.0000
0.0638
0.0012
0.0048
0.0000
0.1060
0.4600
0.4540
0.4900
0.0056
0.0515
0.0000
0.4440
0.2220
0.0000
0.4440
0.0700
0.0000
0.0000
0.0000
0.1370
0.7400
0.2740
0.0137
0.0000
0.0000
6.1500
0.0000
0.0000
0.0000
0.0630
0.0740
0.0000
.21. 0000
0.2320
0.0750
0.2770
0.1570
0.0800
0.1960
"0.0334
0.0028
0.0000
0.0290
0.0004
0.0000
0.0000
0.1170
0.0940
0.0000
0.1111
0.0045
0.0000
0.0000
0.4770
2.5300
2.0900
1.8200
0.0730
0.0000
0.0000
0.8200
0.4730
0.0000
0.5550
0.0008
0.0000
0.0000
0.0000
2.8300
0.5000
3.3300
0.1860
0.1940
0.0000
12.0800
0.0000
0.0000
0.0000
0.1580
0.1502
0.0000
_9.7100
0.1220
0.1000
0.1350
0.1180
0.0680
0.2590
0.0194
0.0121
0.0002
0.0146
0.0001
0.0001
o.oooo"
0.1158
0.0368
0.0006
0.0445
0.0012
0.0010
0.0000
1.2978
1.4398
0.9517
1.1711
0.0448
0.0088
0.0000
0.4049
0.4302
0.0350
0.5171
0.0146
0.0136
0.0000
0.5937
2.0535
1.5534
1.5597
0.0574
0.0027
0.0000
1.0993
0.1955
0.0315
0.3881
0.0259
0.0233
0.0000
3.8610
0.8696
0.2548
1.4743
0.0801
0.0558
0.0508_
0.0135
0.0066
0.0003
0.0139
0.0021
0.0000
o.oooo"
0.0558
0.0285
0.0033
0.0592
0.0080
0.0005
0.0000
1.1656
1.4585
1.0029
2.0245
0.1433
0.1198
0.0000
0.2824
0.1397
0.0006
0.2956
0.0398
0.0023
0.0000
2.8689
1.0870
0.3646
1.9550
0.2541
0.0537
0.0000
1.0989
0.0361
0.1169
0.3199
0.0165
0.0111
0.0000
6.0748
0.1679
0.1521
0.4432
0.0606
0.0315
0.0508_
Clearly this is a reducible case with one obvious Perron eigenvalue X 2 — 0.0508. Applying seven
iterations we find that the other Perron eigenvalue Ai has the positive probability eigenvectors.
(6.5)
Z\R —
0.00052
0.00198
0.30482
0.03033
0.48015
0.03087
0.15134
Z\B —
0.00082
0.00433
0.54771
0.01396
0.26523
0.06485
0.10310
658
where also
D. R. HOWES AND R. M. THRALL
(6.6)
a lfi = 0.85983, a 1B = 1.33191
\i = a R a B = 1.14522
WiR = a XR ZiR, W xb =olibZ
\H.
Since Xi is much greater than K2, the ideal weights obtained from Xi may be regarded as being more
significant than those obtained from ta as given in (6.7) and (6.8) below:
(6.7)
Z2R — Z2B —
(6.8) a 2R = 0.1960, a 2B = 0.2590, A 2 = 0.0508,
WiR — OCzrZzr, W2B — OC2BZ2B.
BIBLIOGRAPHY
[1] Barfoot, C. B., "The AttritionRate Coefficient, Some Comments on Seth Bonder's Paper and a
Suggested Alternative Method," Operations Research 1 7, 888894 (1969).
[2] Bonder, Seth, "A Theory for Weapon System Analysis," Proc. U.S. Army Operations Research
Symposium, 111128 (1965).
[3] Bonder, Seth, "The Lanchester AttritionRate Coefficient," Operations Research 15, 221232
(1967).
[4] Bonder, Seth, "The Mean Lanchester Attrition Rate," Operations Research 18, 179181 (1970).
[5] Bonder, S. and R. Farrell, "Development of Models for Defense Systems Planning," SRL 2147,
Systems Research Laboratory, University of Michigan, Ann Arbor, Michigan (1970).
[6] Corg, "Measuring Combat Effectiveness," Vol. II, Technical Operations Incorporated Inc.
Combat Operations Research Group, Alexandria, Va. (Jan. 1970).
[7] Dolansky, L., "Present State of the Lanchester Theory of Combat," Operations Research 12,
344358 (1964).
[8] Frobenius, Georg, "Uber Matrizen aus nicht negativen Elementen," Sitzungsberichte der Kgl
Preussischen Akademie der Wissenschaften zu Berlin (1912), Berlin, pp. 456477.
[9] Frobenius, Georg, Gesammelte Abhandlungen, Band III (Edited by JP. Serre), SpringerVerlag,
Berlin (1968).
[10] Gantmacher, F. R., The Theory of Matrices (Chelsea, 1959), 2 vols.
[11] Grubbs, Frank E. and John H. Shuford, "A New Formulation of Lanchester Combat Theory,"
Operations Research 21 , 926941 (1973).
IDEAL LINEAR WEIGHTS 659
[12] Hayward, P., "The Measurement of Combat Effectiveness," Operations Research 16, 314323
(1968).
[13] Hero, "Comparative Analyses of Historical Studies," Historical Evaluation and Research Office,
2223 Wisconsin Avenue, Washington, D.C. (15 Oct. 1964), Annex IIIH.
[14] Householder, A., Principles of Numerical Analysis (McGrawHill, New York, 1953).
[15] Kimbleton, S., "Attrition Rates for Weapons with MarkovDependent Fire," Operations Research
19, 698706 (1971).
[16] Koopman, B. O., "A Study of the Logical Basis of Combat Simulation," Operations Research 18,
855882 (1970).
[17] Lanchester, F. W., Aircraft in Warfare, the Dawn of the Fourth Arm (Constable, London, 1916).
[18] Morse, Philip M., and George E. Kimball, Methods of Operations Research (John Wiley, New York,
1951).
[19] Perron, Oskar, "Zur Theorie der Matrices," Mathematische Annalen, Vol. 64 (1907).
[20] RACTPIII, "Tacspiel War Game Procedures and Rules of Play," Research Analysis Corp.
McLean, Va. (Nov. 1963) (Secret).
[21] Rustagi, J. and R. Laitinen, "Moment Estimation in a MarkovDependent Firing Distribution,"
Operations Research 78, 918923 (1970).
[22] Rustagi, J. and R. Srivastava, "Parameter Estimation in a Markov Dependent Firing Distribution,"
Operations Research 16, 12221227 (1968).
[23] Snow, R. N., "Contributions to Lanchester Attrition Theory," Project RAND RA~ 15078 Douglas
Aircraft Co., Santa Monica, Cal. (Apr. 1942).
[24] Shuford, John H., "A New Probability Model for Lanchester's Equations of Combat," Masters
Thesis submitted to the George Washington University (Dec. 1971).
[25] Taylor, James G., "A Note on the Solution to Lanchester Type Equations with Variable Coeffi
cients," Operations Research 19, 709712 (1971).
[26] Thrall, R. M. and Associates, Final Report to U.S. Army Strategy and Tactics Analysis Group,
RMT200R433 (1 May 1972).
[27] Todd, J. (Editor), Survey of Numerical Analysis (McGrawHill, New York, 1962).
[28] United States Army Combat Developments Command Report, Measuring Combat Effectiveness,
by Technical Operations Incorporated, Combat Operations Research Group, Vol. I "Firepower
Potential Methodology (U)" (ConfidentialNO FORN).
[29] Varga, R., Matrix Iterative Analysis (PrenticeHall, Englewood Cliffs, New Jersey, 1962).
[30] Weiss, H. K., "LanchesterType Models of Warfare," Proc. First International Conference on
Operations Research (Dec. 1957), pp. 82  89.
DECISION RULES FOR ATTACKING TARGETS OF OPPORTUNITY
David V. Mastran
OASD(SA)
Department of Defense
and
Clayton J. Thomas
Hq., USAF
ABSTRACT
Frequently in warfare, a force is required to attack a perishable enemy target system — a
target system where the targets are detected seemingly at random, and if not immediately
attacked, will shortly escape from detection. A conflicting situation arises when an attack
element detects a target of relatively low value and has to decide whether to expend his
resources on that particular target or to wait for a more lucrative one, hoping one will be
found. This paper provides a decision rule giving the least valued target that should be at
tacked as well as the resources that should be expended as a function of the attack element's
remaining mission time.
I. INTRODUCTION
Frequently in warfare, a military force is required to attack a perishable target system. A perishable
target system — in contrast to a fixed target system — is one in which the targets are detected seemingly
at random, and if not immediately attacked, shortly escape. These "targets of opportunity" are of
varying military value and the attacker can attack only a limited number. The number of bombs, mis
siles, torpedos, or bullets carried by the attacker and the amount of time available to find, attack, and
destroy the target are the limiting factors. A conflicting situation arises, then, when an attack element
finds a target of relatively low value and has to decide whether to expend resources on that particular
target or to wait for a more lucrative target, hoping one will be found. The items of information neces
sary to make this decision, as well as the means for manipulating the information mathematically,
are the subjects of this paper.
Some hypothetical, yet pertinent, examples of this military situation follow:
1. Army
A combat patrol is waiting in an ambush position for an enemy unit to pass by. The patrol is limited
to executing one ambush since its presence will then be exposed. Because of morale and endurance
problems, however, the patrol can wait for only 3 days by the trail. If one enemy soldier comes down
the trail on the second day, under what conditions should the patrol attack him and forego the oppor
tunity of attacking a larger enemy unit?
2. Air Force
An aircraft or flight of aircraft with limited loiter time and ordnance is sent out to attack enemy
truck convoys on the Ho Chi Minh trail. The convoys present themselves seemingly at random, appear
ing from under the jungle foliage, turning off on hidden roads, and again disappearing. These convoys
661
662 D. V. MASTRAN AND C. J. THOMAS
are of varying sizes consisting of from 1 to 50 or more trucks, and therefore, are of differing values to
the attacker, v^iven knowledge of the expected number of trucks destroyed for the convoy size attacked,
how should the attacker decide which convoys to attack and how much ordnance to expend, as the
remaining loiter time decreases?
3. Navy
A conventional submarine carrying 16 torpedos is on patrol in the North Atlantic assigned to a
shipping lane along which enemy ships are known to travel. The submarine can stay on patrol for a
limited number of days. The targets, ships, vary in value to the submarine commander according to
tonnage (or, perhaps, cargo carried if the nature of the cargo can be determined). The submarine com
mander may leave a detected ship if he believes it is not a lucrative target, or he may delay his decision
to fire torpedos and remain with the ship, thereby foregoing the chance of finding other valued ships.
What size ships should the commander attack and how many torpedos should be expended as a func
tion of the time the submarine has remaining on station?
In these three examples, there are several elements in common. First is that the targets are perish
able (fleeting) and of varying value. Second, the attacker can attack only a relatively small number of
targets. And third, there is a limited time within which the attacker must act. There are, in fact, many
situations both in and out of warfare that have similar conflicting elements in common. The literature
is expanding in the area of opportunity analysis.
II. RECURSIVE RELATIONSHIP: ATTACK ONLY ONE TARGET
The basic approach in solving the problem will be the use of Dynamic Programming. A recursive
relationship will be derived to obtain a decision rule that specifies when to attack targets. For sim
plicity, the assumption will be made initially that each attack element can only attack one target
and that all ordnance will be expended in the attack.
Since the targets are assumed to be detected at random, a probability distribution will be defined
over the times between target detections. From operational data, there may already be available a
probability distribution of target "interdetection" times; otherwise, one will have to be assumed. Ti
denotes the probability that i time intervals separate succeeding detections.
(i) 2 Tt=1 
The conditional probability Dt that there is a target detection in the next time interval given the last
detection occurred i — 1 time intervals ago is:
(2)
and
Di=7y(l2 7}) for i^2
0, = 7V
Let g(v) be the probability density function of target values measured in terms of the expected
target destruction. The distribution is assumed to be stationary and may be discrete for certain classes
of targets. Given a target is detected, then, its value V is a random variable drawn from g(v).
TARGETS OF OPPORTUNITY 663
Let/„(i) be the expected return to the attack element — using an optimum policy — when n time
intervals remain till the mission is over and i time intervals have passed since the last target detection.
The decision rule is of the form: "Attack the target when n periods remain if and only if its value
V is greater than K n " K n is a variable changing over time that describes the least valued target that
can be attacked.
Given a target detection, then, the expected return of the attack, for the case in which g{v) is
continuous, is
(3)
/«i(l) f g(v)dv+f vg(v)dv.
J »««„ J v&K n
If no target is detected in the rath remaining time interval, the situation transforms to the state
with n — 1 time intervals remaining and i+1 intervals since the last detection, i.e.,/ n i(i+ 1).
Combining the terms and the detection probabilities, we get
(4)
f n (i) = Max\ D t \ f fni(l)g(v)dv+( vg(v)dv] + (lD i ) [/«l(i+l)]l
Obviously f n {i) is maximized only when a target is attacked that has a higher value than could
be obtained by waiting another time interval and obtaining/„_i(l). Hence K n =fni(l). Thus
(5) Mi)=Di\ f f n  x {\)g{v)dv+[ vg{vydv\ + 0.Di) [/_i(i+l)] f
(6) Mi) = Dif vg(v)dv, and/ o (i) = 0.
Jv»o
By using (5) and (6), we can find/ n (i) for all n and i.
EXAMPLE 1.
Consider the case of an A — 26 aircraft with 2 hours of loiter time being sent out to attack convoys in
Laos. For simplicity, the target interdetection time distribution will be assumed to be geometric with
a mean of 10 minutes between detections.
D l = D 2 = . . . =D X = 0.1
Additionally, because of the memoryless property of the geometric distribution:
(7) Mi) =Mi+l) alH^l.
Table 1 shows the expected return V in terms of trucks destroyed or damaged given the convoy
size attacked. Also provided is the probability distribution of target values (or convoy sizes), g(v),
which are discrete in this example.
664
D. V. MASTRAN AND C. J. THOMAS
Table 1
Convoy size=i
Vi
g(v) = P r {V=V l }
1
0.25
0.15
2
0.40
0.17
3
0.55
0.20
4
0.70
0.15
5
0.85
0.10
6
1.00
0.05
7
1.15
0.04
8
1.30
0.03
10
1.60
0.03
15
2.00
0.03
20
2.15
0.03
30
2.25
0.02
Some convoy sizes are missing from the table, not necessarily because the enemy operates only
in the convoy sizes shown above, but, in part, because the values of £ that are given suffice to represent
those that are perceived and reported. The recursive relationship in this discrete example is,
(8)
fn = 0.1
2 /»•*(») + 2 *#(*)1+  9 /
The solution is shown in Table 2.
/o() = 0.
Table 2
Loiter time remaining= n
(min)
Jn A.n+1
Minimum size
convoy can attack
10
0.505
3
20
0.749
5
30
0.915
6
40
1.043
7
50
1.149
7
60
1.239
8
70
1.317
8
80
1.386
8
90
1.448
8
100
1.503
8
110
1.553
8
120
1.597
8
Thus, the expected return for the A26 with 2 hours of loiter using an optimal policy is 1.597 trucks
damaged or destroyed. This compares quite favorably with 0.758 trucks damaged or destroyed, which
is what the A26 would expect to achieve by attacking the first convoy detected.
A possible objection to this decision rule is that the enemy would observe that only the large con
voys were being attacked and would begin operating in smaller convoys. The rule, however, allows
even a onetruck convoy to be attacked provided the remaining loiter time is short enough. Moreover,
this change in enemy policy could be expressed in a nevf g(v).
TARGETS OF OPPORTUNITY
665
EXAMPLE 2.
In this extension of Example 1, the dependence of/ n (t) on i — the time since the last target detec
tion—is examined for two different distributions. All values are the same as in Example 1 except for
the interdetection time distribution. Let distribution A be defined by
(9)
r,=o.o6,
7=1,2, . .
.,15
Tj=0,
7= 16, . .
.,27
7j=0.1,
j=28
7^=0,
j=29, 30 .
. .
and distribution B by
(10)
r,= o.o9i,
7^ = 0.001,
Tj=0
7=1,2, . . .,10
7=11, 12, . . ., 100
7=101, 102, . . .
B.
Distributions A and B, though quite different from each other, and very different from the geometric
distribution of Example 1, have the same mean value as that of Example 1, namely 10. That is:
(11)
An interesting question is how much larger or smaller than/„(l) does/ n (£) become as i is varied.
For representative values of n, and the two distributions A and B, Table 3 provides the answer.
Table 3
R
Distribution A
Distribution B
min/„(i)
/ n (D
max/„(£)
min/„(i)
/(I)
max/„(i)
10
20
30
40
50
60
70
80
90
100
110
120
0.000
0.745
0.925
1.000
1.143
1.240
1.320
1.388
1.451
1.508
1.558
1.603
0.458
0.760
0.949
1.063
1.170
1.261
1.338
1.406
1.467
1.522
1.570
1.614
0.784
0.939
1.074
1.164
1.255
1.334
1.402
1.463
1.518
1.567
1.611
1.651
0.087
0.195
0.315
0.443
0.577
0.717
0.862
1.013
1.170
1.246
1.308
1.362
0.716
0.890
0.998
1.075
1.140
1.198
1.256
1.315
1.376
1.441
1.499
1.548
0.873
1.027
1.114
1.179
1.232
1.282
1.332
1.382
1.436
1.494
1.546
1.591
From Table 3, the sensitivity of f n (i) to variations in i is much greater for small n. Just how large
n must be before the variations become negligible depends on the distribution. By comparing the
/„(1) with the/„() of the geometric distribution in Example 1, one sees the results are moderately
close despite the large differences in the distributions.
QQQ D. V. MASTRAN AND C. J. THOMAS
The decision rule formulations may have applications other than that of making real time deci
sions. If, for instance, the question were raised — "What would more loiter time allow in terms of in
creasing expected trucks damaged or destroyed per sortie?" — an answer can be given directly by
plotting /„(i) versus n. The loiter time of an aircraft can be substantially increased by adding sufficient
tanker support or decreasing the ordnance load. By assessing the costs of extending the loiter time with
the gains achieved, an "optimum" loiter time can be derived.
The example presumed that the decision making was decentralized; the individual aircraft com
mander made the decision to attack. The attack element in general, however, is really part of a larger
force of attack elements, each being assigned a specific sector to patrol. If the decisionmaking func
tion were centralized, information on the level of enemy activity could be gathered through several
intelligence sources and a better perspective attained on the chances of detecting more valuable targets.
The centralized decisionmaking function would not necessarily dictate to the attack elements where
to look for targets within a particular sector, but once a target was detected, would give a gonogo
decision to attack.
III. RECURSIVE RELATIONSHIP: MULTIPLE TARGET ATTACKS POSSIBLE
The recursive relation of the last section is generalized in this section to allow more than one attack.
Let/ n (i, rn) be the expected return tor the attack element using an optimum policy when i time units
have elapsed since the last target detection and m ordnance units remain to be expended. If a target
of value V is attacked with a salvo of W units of ordnance, h(V, W) will be the expected return. In
general, the function h will satisfy several inequalities. For the case where h is differentiable in V,
these may be written as:
(12) h(v, w) <h(v, w+1)
dh(v, w)
dv
>0
dh(v, w— 1) dh{v,w)
dv dv
More precisely, h(v, w) is a monotonically increasing function in both v and w with the property that
the rate of increase of the expected return increases with increasing w.
With only one time unit left in the mission, the attack element will attack any target and expend
all ordnance. Thus,
(13) Mi,W)=Dit h{v,W)g{v)dv,
Jail v
where D t is the conditional probability of detection in the next time interval, given the last detection
occurred i — 1 intervals ago.
The decision rule for this formulation of the problem is: "If, when n time units remain in the
mission and m ordnance units are available, the value of the target detected V falls in the half open
interval [K* m , Kg^ 1 ) , then use W units of ordnance in an attack on the target."
TARGETS OF OPPORTUNITY 667
By definition K% t m = and K™+£ = » for all n and m. Notice that the constants K% t m depend both on
the amount of ordnance available and on the time remaining in the mission. For notational simplicity,
however, the subscripts n and m will be omitted when they do not change in the formulations presented.
Assuming an attack takes one unit of time, the recursive relationship is
(14)
fn(i,m) = max \ D t Y [h(v, w) +/"„_, (1, mw)]g(v)dv+ (1 D i )f„ 1 (i+ 1, m) ,
* I j£o Jk w )
where K is the set of m + 2 tuples
K={(K°,K\ . . ., K m+1 ) : «s K" < °° and K a <K a + l , a = 0, 1, . . ., m} and/„=0 for n <0.
To maximize /„(i, m) over the set of m + 2 tuples, differential calculus can be used. f n \(i, m), g(v),
and D< are independent of K^ t m = K w .
Selecting any element of the m + 2 tuple and taking partial derivatives,
(15)
d/ f^ a m) =^a D i \ K [_S h ^ al)+fni{h ma+l)]g(v)dv
+ W aDi \ a [*(»,a)+Ai(l, ma)]g(v)dv
or
(16) dfn d % m) = D ig (K°)A,
where
(17) A = h(K a , a\)+f n i{l,ma+l)h(K a , a) /„_,(1, ma).
Since g(K a ) > 0, /4 = at a minimum or maximum. To test for a maximum, one calculates
(18) »X^. D < (K . ) M, +Dl U!glA.
Since the second term becomes at A = 0, one has
(19) 3 ^ a , "#1* ; ^ dK „ dKa J
The term in brackets is negative because of the properties postulated for h(v, w). Therefore, when
A =0, one has the maximum value of f n {i, m). K a , then, using (17) is the solution to the equation
668 D v  MASTRAN AND C. J. THOMAS
(20) h(K",al)h(K", a) =/„,(!, ma) /„_,(!, ma+1).
EXAMPLE 3.
For this example, assume a submarine has four torpedos remaining and 100 hours left on station.
The single shot hit probability p is assumed to be independent of target value and equal to 0.3. *
The expected return, given that the target value is V and the number of torpedos fired is W, is given
by
(21)
h(v,w)=v[i{i P n
For simplicity, the interdetection time distribution will be assumed geometric with a mean of
20 hr:/), = 0.05, i=l,2, . . ., «.
The normalized target values will be assumed to be distributed uniformly between and 10.
(22)
g(v) = Ul0, O^v^lO
= 0,
otherwise.
The equation yielding K a , after introducing the functional form of (21), becomes
(23)
K*=\fni(l, ma+1) fniO., ma)]lp(lp) a K
The solution procedure is first to find K'{ m for all m and a given f=0. Then having the K'( m find the
/i(l, m). This procedure is repeated, first finding the K% t m> then the/„(l, m).
The solution is shown below in Figures 14. Figure 5 represents the situation in which the sub
marine has m torpedos, nt=l, 4. Along the ordinate is target value and along the abscissa the time
remaining on station. The numbers 04 indicate the number of torpedos that should be fired in the
specific "regions" of target value and time remaining on station.
m  4
20
40 60
mission time remaining
Figure 1
*In general, p does not have to be independent of target value.
80
100
TARGETS OF OPPORTUNITY
669
m = 3
20
40 60
MISSION TIME REMAINING
Figure 2
80
100
m = 2
20
40 60
MISSION TIME REMAINING
Figure 3
20
40 60
MISSION TIME REMAINING
Figure 4
80
100
If the submarine attacked the first target found with all four torpedos, the expected return would
be 3.78. On the other hand, by employing the decision rule just described, the submarine can expect
to obtain a return of 6.72, a 78percent improvement.
670 D v  MASTRAN AND C. J. THOMAS
EXAMPLE 4.
In the previous example, the submarine commander was assumed to salvo all torpedos. If the com
mander desired to employ a "shootlookshoot" policy, the decision rule and the formulation must be
modified to permit the expenditure of less than w units of ordnance, when observations of prior damage
so suggest.
Assuming p is the single shot probability of hit, the expected return for a "shootlookshoot"
policy — given W units of ordnance can be expended and a target of value V is detected — is
(24) p[V+f n i(l,ml)] + p(lp)[V+fni(l,m2)]
+ . . .+p(lp) w  1 [V+f n i(l,mW)] + (lp)*f n  1 (l,mW).
The last term is included since f n ( 1 , rn — W) is returned whether or not the last ordnance unit hit the
target. Simplifying the above expression, the return, expressed as h' (V, W) is
(25) [l(lp)»]V+ ^ p{\ ~ pVy n i{\,m  j) + {\  p^fn.^l^ W).
The recursive relationship is
0,t h'(v,w)g{v)dv+(lD,)f n i(i+hm)\.
Finding the K a as before by taking partial derivatives,
df„(i,m)
dK a
where
Dig(K°) A,
(27) A=p(lp)iK°+(lp)''ifni(l,ma+l)  (1 p) a '/ n _,(l, m  a).
Setting A — for the maximum yields
^ m = [fnt(l,ma+l)fni(l,ma)]lp.
n,m
The solution for this formulation produces two somewhat surprising results. First is that K a n m
= Kn7mi f° r au " n and * m an( ^ a> 1 Instead of four figures as in the last example, then, only one
need be drawn (Figure 5), since the uppermost region signifies using up to all remaining ordnance units;
the next uppermost region, all but one remaining unit of ordnance, etc.
TARGETS OF OPPORTUNITY
p = 0.3
671
40 60
MISSION TIME REMAINING
80
100
Figure 5
The second surprising result is that, K™ m is independent of p, the single shot probability of hit;
the other values of K" m for a ^ m are dependent on p, however. In fact, as p — > 0, K 1 * K m
for all n and m; in effect saying either expend no ordnance or expend up to all remaining ordnance. On
the other hand, as p — > 1, K n , m ~* making the decision rule more selective in establishing the number
of ordnance units to expend. Figure 6 shows the results for p = 0.99.
p = 099
20
80
100
40 60
MISSION TIME REMAINING
Figure 6
In terms of the improvement in expected return for a "shootlookshoot" policy, the submarine
commander with 100 hours remaining can expect a total return of 7.65 versus 6.72 for the salvo decision
rule. This represents 13.8percent improvement over the salvo rule and a 102percent improvement
over the expected return when salvoing all four torpedos at the first ship detected.
EXAMPLE 5.
In this simple extension to Examples 3 and 4, the target values or expected return will be allowed
to change over time. If the mission time is long and the war is expected to be a short intense one, the
same target will be "worth" more the earlier it is destroyed. Allowing the expected return to change
can be accomplished by discounting the target values as the mission time decreases.
Let
h(v, w\n) =h(v, w) • (1 —a"),
672
D. V. MASTRAN AND C. J. THOMAS
where (I — a") is the discount term.
Because of the structure of the recursive relationship, the discount term can be factored out of the
expression leaving the previous results valid for determining K a n m .
(28)
f m f K w+1 1
/»(», m)= (la«) max \D, ;Y h(v, w)g(v)dv + (1 D i )f„ 1 (i + 1. m).
By letting a = 0.95, the table below compares K£ 4 from Examples 4 and 5.
Table 4
n
K* i4 (Example 4)
K* „ (Example 5)
10
1.872
0.431
20
3.266
1.330
30
4.249
2.287
40
4.981
3.163
50
5.546
3.926
60
5.997
4.578
70
6.364
5.132
80
6.670
5.602
90
6.928
6.002
100
7.148
6.343
Notice that by discounting, the decision rule becomes more lenient in allowing attacks using all
ordnance. This occurs because the target values are decreasing.
There are other extensions to the formulations than can be incorporated such as allowing the
attack time to exceed a unit time interval or changing the detection probabilities after an attack has
been executed. These extensions are simple ones and hence will not be addressed. The flexibility of
the Dynamic Programming approach is considerable.
IV. CONCLUSIONS
The Dynamic Programming formulations appear to be quite useful in determining what a lucrative
target is. To the extent that the distributions required can be estimated or empirically derived, the
approach proposed in this paper will be valid. Other considerations such as the threat to the attack
element from AntiAircraft Artillery, AntiSubmarine Warfare, etc., must be addressed, however.
Fut»»*» work will investigate the effects of attrition on the decision rule.
BIBLIOGRAPHY
[1] Bellman, Richard E., Dynamic Programming (Princeton University Press, Princeton, New Jersey,
1957).
[2] Bellman, Richard E. and Stuart E. Dreyfus, Applied Dynamic Programming (Princeton University
Press, Princeton, New Jersey, 1962).
TARGET SELECTION IN LANCHESTER COMBAT: LINEARLAW
ATTRITION PROCESS
James G. Taylor*
Department of Operations Research and Administrative Sciences
Naval Postgraduate School
Monterey, California
ABSTRACT
We develop the solution to a simple problem of target selection in Lanchester combat
against two enemy force types each of which undergoes a "linearlaw" attrition process.
In addition to the Pontryagin maximum principle, the theory of singular extremals is required
to solve this problem. Our major contribution is to show how to synthesize the optimal
target selection policies from the basic optimality conditions. This solution synthesis
methodology is applicable to more general dynamic (tactical) allocation problems. For
constant attritionrate coefficients we show that whether or not changes can occur in target
priorities depends solely on how survivors are valued and is independent of the type of
attrition process.
1. INTRODUCTIOIN
In a recent paper [26] we have presented some elements of a mathematical theory of target selection
in dynamic combat situations. We did this through the examination of the structure of the optimal
allocation policies for some tactical situations described by Lanchestertype equations of warfare.
The purpose of this previous paper was to contrast the structures of the optimal allocation policies
for various scenarios. In the present paper we develop results for a prescribed duration battle in which
enemy target types undergo a "linearlaw" attrition process (see section 3). For reasons of*brevity, we
had previously [26] just stated these results without justification.
The problem under study is solved by the mathematical theory of optimal control. Its solution,
however, requires more than the wellknown Pontryagin maximum principle [22]: the theory of singular
extremals (see chapter 8 of [4] ) must be used to solve it. A brief discussion of the required theory of
singular extremals is included in this paper. By an extremal we mean a battle trajectory on which the
necessary conditions of optimality are satisfied everywhere in time.
The major contribution of this paper is to show how to synthesize the optimal control in combat
against target types which undergo a "linearlaw" attrition process. In this case, singular subarcs (see
section 2) may be present in the battle trajectory. By the synthesis of optimal control, we mean the
explicit determination of the time history of the optimal control from initial to terminal time as a func
tion of the initial state of the system. There is no general method for the synthesis of optimal controls
in singular problems [19]; each class of problems possesses its own peculiarities. Hence, an under
tanding of how to synthesize the optimal control in this elementary problem is particularly important,
*This research was supported by the Office of Naval Research as part of the Foundation Research Program at the Naval
Postgraduate School and partially under Project Order No. 20150.
673
674 J G. TAYLOR
since it provides insight tor more complex extensions that we have considered in our subsequent
researches.
The body of this paper is organized in the following fashion. First, we review that part of the
theory of singular extremals which is required for the solution of the problem under study. Next, we
present our model and develop the basic necessary conditions of optimality. Then, we discuss how to
synthesize extremals using the necessary conditions. Next, we explain how the optimality of the
extremal control is demonstrated by the existence of an optimal control and the uniqueness of extremals.
Then, we show how to synthesize the solution to our problem. This is done for the two cases of import.
Finally, we make some comments about the structure of the optimal target selection policies and
extensions.
2. THE THEORY OF SINGULAR EXTREMALS
In an optimal control problem, the maximum principle may fail to determine an optimal trajectory,
since the maximization of the Hamiltonian may not lead to a welldefined expression for optimal control
[16], [14] (also see chapter 8 of [4]). Singular solutions usually occur when the Hamiltonian is a linear
function of the control variables. However, all problems for which the Hamiltonian is a linear function
of the control variables do not have singular subarcs in their solution.
The problem that we shall consider has one control variable, and it appears linearly in the
Hamiltonian. By a singular subarc we denote that part of an optimal trajectory on which the maximum
principle cannot be used to determine the control because the coefficient of the control variable in the
Hamiltonian is zero* (see pp. 226227 of [15]). Then the term "singular solution" will be used to refer
to any optimal trajectory which contains one or more singular subarcs.
To elaborate further, when the Hamiltonian H is a linear function of the control variable <£, then if
——=0 for a finite interval of time (or, another way to say this, the coefficient of <b vanishes identically
d<p
for a finite interval of time), then the maximum principle does not determine the control. Observe that
in this case all feasible values of </> maximize the Hamiltonian. When this happens we determine the sin
, dfl dH
gular control by requiring that we remain on the singular subarc, i.e.,—— remains zero. If — is to be
d<p d<p
identically equal to zero for a finite interval of time, then all its derivatives with respect to time must
also be equal to zero. We determine the singular control, which keeps the system on the singular sub
arc, by considering as many of the time derivatives of— as are required for the control variable <b to
d<b
appear explicitly so that it may be determined from an algebraic equation. Thus, in general we consider
(1) = — = (— \=— (— \
d<f>~ dt\d<j>) ~ dt 2 \d<b) ~
For the problem at hand, an explicit expression for the singular control is obtained from the equation
(2) <fe»W/~°
This statement, of course, depends upon the form of Hamiltonian employed (see section 5).
LANCHESTER TARGET SELECTION 675
dH d /dH\
by use of the conditions — = — (~r7l~0 ano " tne canonical equations (i.e., both state and adjoint
a<p at \d<p/
system).
We must further check to make sure that we can get a maximum return (in the case when we wish to
maximize the criterion functional) from use of the candidate singular subarc. The following condition
(generalized LegendreClebsch condition) is necessary for a singular subarc to yield a maximum return
d f d 2k /dH\)
It is obtained by examining the negative semidefiniteness of the second variation for a special class of
explicitly defined control variations [17]. For the problem at hand, it suffices to consider the generalized
LegendreClebsch condition with k~l. Recently, Jacobson [10] discovered a new necessary condition
for optimality on singular subarcs. This condition is not readily checked, however, for the problem at
hand, since the details of application are extremely messy.
It should be emphasized that the generalized LegendreClebsch condition (3) is merely a necessary
condition of optimality so that even if it is satisfied on a singular subarc, we are not guaranteed that
the criterion functional is maximized. Very recently, Jacobson [11] gave sufficient conditions for non
negativity of the second variation in singular and nonsingular control problems. These yield conditions
analogous to the wellknown noconjugatepoint condition (see pp. 181184 of [4]) for singular subarcs
and have led to necessary and sufficient conditions of optimality for singular control problems [23]
(see also [13]). In this paper we will, however, use a different approach to prove that we have found
the optimal control (see section 7 below). The interested reader can find comprehensive bibliographies
on the singular control problem in [11], [12], and [17].
3. THE MODEL
We consider the following prescribed duration battle
subject to:
maximize {ry(T) —pxi(T) —qx 2 (T)} with T t specified,
d%\ ,
dx2 n j.\
(4) jT = — &1*1 — &2*2,
with initial conditions
xi(t = 0)=x° 1 , x 2 (t = 0)=x° 2 , y(t = 0)=y ,
and
xi, x 2 ,y^0, ^ (f> *£ 1, and T *£ T u
676 J G. TAYLOR
where all symbols are defined in section 4 and the battle termination conditions are elaborated upon
below.
The battle lasts for *£ t ^ T\ unless, of course, one side or the other is annihilated before T\. To
be more precise, the battle terminates under one of the following three conditions:
TC 1 ±x l (T=x 2 (T)=0 and T < T u
(5) TC 2 :y(T)=0 and T ^ T u
TCy.T=T u
where T denotes the time at which the battle ends.
In the above problem (4) xi, x 2 , and y are called state variables, while <f) is called a control (or
decision) variable. A constraint, such as x\ 3* is called a state variable inequality constraint (SVIC)
and requires special treatment (see chapter 6 of [22]). In other words, the wellknown maximum prin
ciple* (as presented in chapters 13 of [22]) requires modification in problems with SVIC's (see also
chapter 3 of [4]). Moreover, Mclntyre and Paiewonsky [21] remarked in 1967 that "the optimal control
problem with state space constraints does not appear to be well understood." It has been the personal
experience of the author that this is even true in the applications literature today [24].
Using the corner conditions from the theory of state variable inequality constraints (SVIC's) (see
[14], [21], or pp. 125 126 of [4] ), it will be shown below that an optimal policy can only lead to the follow
ing extremal terminal states
£, : x 1 (T) = x 2 (T) = 0, y(T) > 0, andT 1 ^ 7\ with*i(0 > Ofor* < Tandx 2 (t) > Oforf <7\
(6)E 2 :xi(T) > 0,x 2 (T) >Q,y(T) = 0, andr*£ 7\,
E 3 :x x (T) >0,x 2 (T) >0,y(T) > 0, andT=Tu
In other words, for t < T we have x*(t) > 0, x*(t) > 0, and y*(t) > so that no SVIC is active for
t < T when an optimal policy is followed. Hence, the SVIC's are essentially ignored in developing the
solution to the problem at hand (except, of course, for the establishment of the fact that for an optimal
policy* *(t) > Owhenf < T, etc.).
In previous papers [26], [27] we have described the basic scenario under consideration and also the
circumstances which lead to a "linearlaw" attrition process. As before [26], we refer to attrition as
being a "linearlaw" process when the casualty rate is proportional to the product of the number of
enemy firers and remaining targets. We have also discussed at length the structure of the optimal target
selection policies and its implications for military tactics previously [26].
Since the state and adjoint equations do not readily yield an analytic solution for quantities such
as Xi(t) or pa(t), it has not been possible to obtain explicit expressions for certain model parameters.
* There is a difference in sign between the version of the maximum principle used by Pontryagin et al [22] and an equivalent
version commonly used in the control theory literature of this country (see p. 108 of [4] ).
LANCHESTER TARGET SELECTION 677
Moreover, the author has not been able to derive explicit expressions for optimal trajectories or controls
in all cases. However, one can still discuss all the qualitative characteristics of the optimal allocation
policy.
4. NOTATION
The symbols which are used in this paper are defined as follows:
a\,Oi, b\, 62 = constant attritionrate coefficients,
Ei for i= 1, 2, 3 = the tth extremal terminal state as defined by (6)
H = Hamiltonian function,
J = criterion functional = ry(T) — px\{T) — qx2(T) ,
k = constant of proportionality,
L = singular "surface" defined by a\b\X\ = 0262X2,
L' = line (with equation a\px\ — a^qx^) in description of solution to problem,
L= L(<f>, \) = Lagrangian function,
P(t = T) = (x 1 (t=T),x 2 (t=T)),
p, q, r = utilities assigned per unit of surviving X\, X 2 , and Y forces, respectively,
Pi(t) for i— 1, 2, 3= dual variable corresponding \.o\Xiif) {vt\lhxz(t) = y(t) ),
t = time after beginning of battle,
t c — time of occurrence of "corner,"
t e = time of entry to constrained subarc,
ti — T — T/ = time which separates Phase I of the battle from Phase II as described in section 6,
t s = time of entry to singular subarc,
ti = T — Ti = time of first switch in extremal tactics,
T= total time for the battle,
T\= maximum possible duration for battle, i.e., T^Ti,
TCi for i= 1, 2, 3 = battle termination conditions as defined by (5),
v(t) = a t xi (— pi (t) ) — a 2 ac 2 (— Pz{t) ) ,
W— Bellman's optimal value function,
x\, X2, y— combatant force levels; with initial values x?, xi, yo,
8= a positive constant,
81 = a positive constant,
X= a positive constant,
A= Lagrange multiplier defined in (A. 15),
<£ = fraction of Ffire directed at Xi ,
T= "backwards time" from the end of the battle; defined byr= T—t, i.e., the time remaining before
the end of battle,
T ,= T—ti= "backwards time" which separates Phase I of the battle from Phase II as described in
section 6.,
* 1 ,t 2 , etc. = "backwards time" of the first, second, etc., switch in extremal tactics
5. DEVELOPMENT OF BASIC NECESSARY CONDITIONS OF OPTIMALITY
We now develop the basic necessary conditions of optimality which hold on extremals. In appen
dices A and B it is shown that it is nonoptimal to have *i (t ) = with x 2 (t) > 0. Considering the battle
678 J G. TAYLOR
termination conditions (5), it suffices to consider here only the case in which xi(t) > 0, x 2 (t) > 0, and
y(t) > for t < T. Under these circumstances the Hamiltonian for the above problem (4) is given by [4]
(7) H(t, xt, pi, (f>)= (—pia i x 1 y+ p2(i2X2y)4>+ {—P2a 2 x 2 y— p 3 (biXi + b 2 x 2 )} ,
where pi(t) for i = 1, 2, 3 are the dual variables corresponding to the state variables Xi, x 2 , x 3 = y (see
[26], [27] for a discussion of the military significance of these variables). The maximum principle leads
to the following nonsingular optimal control (when there is only one extremal (see section 7.)
r
(0 ior p 2 a 2 x 2 <piaiXi,
1 for p 2 a 2 x 2 > p\a,iX\.
The adjoint system of differential equations for the dual variables is given by
—  = <f>*a l yp l + bip 3 ,
at
(9) ^r=(l<!>*)a 2 yp 2 + b 2 p 3 ,
at
dp 3
— —=<f>*a l x l p 1 + (l—<f>*)a 2 x 2 p2.
at
The boundary conditions for the dual variables at t = T are discussed below.
Additionally, at a corner which occurs at an interior point of the state space (i.e., x\ > 0, x 2 > 0,
y > 0) the following wellknown corner conditions must hold [4]
(10) pdtc) = Pi(tt) fori =1,2, 3,
and
(11) H*(tc) = H*(tt),
where t~ c denotes the time just before the corner, t + c denotes the time just after the corner, and H*(C C )
denotes H(t=tc,xf, pt, </>*). The reader should recall that a "corner" is a place of discontinuity of the
slope of the state space trajectory. Thus, a corner is a place where
02) # «)§<*>.
where
fxt(t)\
x(t) =\x 2 (t) .
\ y(t)J
LANCHESTER TARGET SELECTION 679
a if
On a singular subarc — = for a finite interval of time. From (7) we readily compute that
(13) jt = y{p2a 2 x 2 — Pid x xi ) ,
(14) ~dt\dd>) = ~ ^ xXl + ^2*2) (Pza&z — Piaai) + p 3 y(a 2 b 2 x 2 — ai6i^c,),
and
J 2 / Aff\ rl
(15) rz \TJ) = ~ (P2CI2X2 — piaixi) r (b\X\ + b 2 x 2 ) — 2p 3 (6i*i + b 2 x 2 )(a 2 b2X 2 — 0161*1)
+ y(a 2 b 2 x 2 — aibiXi) ~j~ + Y 2 P3{<f>[ai(a l biXi) + 02(0262*2)] — 02(0262*2)},
where both the state equations (4) and the adjoint equations (9) have been used in the development of
dH
(14) and (15). Considering (13), the requirement that 77 = yields the first condition for a singular
o<p
subarc
(16) Piaai = p 2 aix 2 .
Considering (14) and (16), the requirement that ~r [~ZT ) = on a subarc on which — = yields the
second condition for a singular subarc
(17) 0161*1 = 0262*2.
On a subarc on which the first and second conditions for a singular subarc hold (i.e., (16) and (17)),
we additionally require that
so that evaluating (15) on a subarc on which both (16) and (17) hold we obtain
(19) — (?—) = y*p 3 ai&,*i{0* (ai + a 2 )  a 2 } = 0,
dt 2 \a<p/
which readily yields the singular control which is required to keep the system on the singular subarc
(20) <*>* = spFS.
To check the generalized LegendreClebsch condition on the singular subarc, we further differentiate
680
J. G. TAYLOR
(15) with respect to <f> and evaluate the result on the subarc on which (16) and (17) hold. Doing this, we
find that
(21)
YAi^i)) =y ' p ' it)l(a ' vb ' x,+M ' b ' Xi)>a 
since as is shown below p*{t) > for all t < T. Thus, the necessary condition is met for the singular
path to be optimal.
Considering the battle termination conditions (5), the fact that an optimal policy results injti,x 2 ,
y > for t < T yields that an optimal policy can only lead to the extremal terminal states (6). For each
of these, the boundary conditions for the dual variables at t — T are given in Table I. These results
were obtained as follows. When x\{T) > 0, x 2 (T) > 0, or y(T) > 0, the determination of the value of
the corresponding dual variable at t — T is routine (see [4]). When y{T) = with T < 7\, consideration
of (7) and the transversality condition
(22)
HU = T,xf, Pi ,<p*)=0,
yields that
(23)
p 3 (t=T) = 0.
When Xi(T) = xz(T) = with T < T\ , it is clear by (8) and (20) that we must be on a singular subarc for
T— y =£*=£ T (where y >0). Hence, by (16) and (17) we have that for t<T (since xi(t) and xdf) > for
t<T)
(24)
Pi(t) bi
^7T = T iorTy^t^T.
p 2 («) b 2
Table I. Boundary Conditions of Dual Variables for Various Extremal Terminal States
Terminal
state
E t
:x 1 (T)=x»(T)=0, y(T)>0,
for t < T and x 2 (t) > for t < T.
p l (t=T)=kb l ,
Pi(t=T)=—kb 2 ,
p*(t=T) = r with*>0.
and T
*= 7\ with Xi
(0>o
Terminal
state
E 2
:Xl (T)>0, x 2 (T)>0,
p,(f=7')=p,
P2(t = T)=q,
Pi (t = T) = 0.
y(T) =
= 0,
and r«r,.
Terminal
state
E,
:Xl (T)>0, xz(T)>0,
Pl (t=T)=p,
P2(t=T)=q,
p 3 (t = r) = r.
y(T) > 0,
and T=T\.
LANCHESTER TARGET SELECTION 681
Thus
(25) lim Pi (t)=kbi for i= 1,2.
Since T is unspecified, we also have
(26) H(t,xt,pu<f>*) = 0,
so that on the singular subarc on which (16), (17), and (20) hold, we have that
(27) H(t, xt, Pi, <f>*)=pia 1 x l yp 3 (bix l + b 2 X2).
Combination of (17), (26), and (27) then yields that
(28) ftW =(^W 1
l&i(ai + a 2 )J
Now recalling that for y(T) > we have p 3 {t = T) = r > 0, by continuity of p 3 we obtain thatp 3 (0 >
for t close to T. Hence, by (28) we have
(29) p, (t ) < for all t close to T,
since the quantity in brackets in (28) is positive. By (25) and (29), we have
(30) Pi ( t = T)=kbi for i=l, 2, with k >0.
To establish the fact that p 3 (t) > for all t < T we proceed as follows. It is wellknown [2], [18],
[22] that one can make the identification
(31) p 3 (t)='
dy(t) '
except for certain manifolds of discontinuity (switching surfaces) [2] (see also [3] ) where W is not dif
ferentiate (see p. 73 of [22]). In (31) W = W{t, Xi, jc 2 , y) denotes Bellman's optimal value function [1]
(i.e. W(t, X\, X2, y) denotes the return o tained when the system state aU is (xi, x%, y) and an optimal
policy <f)*(s) is followed for t ^ 5 =£ T). It is wellknown that the corners of broken extremals lie on
such manifolds of discontinuity [2], [3]. Hence, at a point of the state space which lies on an extremal
not at a corner, the W function is differentiable, and we can make the identification (31). Furthermore,
considering the state equations (4), at such a point it is clear that we must have
(32) nit) > 0,
since addition of another Y combatant at t can only result in an increase in return to Y when an optimal
682 J. G. TAYLOR
policy </>*(«) is followed for t =S s *£ T. At a corner we can invoke (10) to establish (32) everywhere for
t< T.
6. SYNTHESIS OF EXTREMAL CONTROLS
By the synthesis of the extremal control we mean the explicit determination of the time history
of the extremal control from initial to terminal time. This is done by combining the extremal control
with integration of the state and adjoint systems of equations. The maximum principle has yielded the
nonsingular control (8) while (1) has yielded the singular control (20) as well as the first and second
conditions for a singular subarc, (16) and (17). It should be noted that this extremal control (both non
singular and also singular) is a function of both the state and adjoint variables.
The above yields a two point boundary value problem: the state and adjoint systems of equations
(i.e., (4) and (9)) are to be solved using the extremal control with (initial) conditions for the state variables
given at t = and (boundary) conditions given for the dual variables at t = T (see Table I). Additionally,
we have established in section 5 that there are only three extremal terminal states. For each of these,
we may start at the end of the battle at t = T (where boundary values for the dual variables are known)
and obtain the time history of the dual variables by a backwards integration of the adjoint system of
differential equations (and also state equations) combined with the extremal control. The corner
conditions (10) and (11) are also used in doing this.
By this process we can trace extremals backwards from each terminal state of battle. This back
wards synthesis process (see also [7] and [9] for further discussions) is carried out so that an extremal
path leads from the terminal conditions to the initial conditions for the case under study. In other
words, the backwards synthesis is carried out in such a way as to guarantee satisfaction of the initial
conditions.
In synthesizing the extremal course of battle (backwards from the end of the prescribed duration
battle) it is convenient to introduce
(33) v(t) = — a t piXi + CI2P2X2.
By (8) and (20) the extremal control may be expressed in terms of v(t) as
, 1 for v(t) > 0,
(34) VU) = \ U% forv(0=0,
Oi + 02
1 for v(t) < 0.
We recall that (17) must also hold on a singular subarc.
Since we develop the solution to this problem by working backwards from the end t — T, it is
convenient to introduce the "backwards time" variable t defined by T — T — t. Observing that
d d
j — — ~j and using both the state equations (4) and the adjoint system (9), we obtain from differentiation
of (33) that
, dv
w 5 ) r= (aibiXi — a 2 bzx 2 )p3
LANCHESTER TARGET SELECTION
683
dv
Thus, we see that on a singular subarc on which v(t) =0we also have that r = 0. Also, it is sometimes
dr
convenient to write (33) as
(36)
v(t)
m
*/ Pi(t) \
Wt)/
Gi)
(a,\b\Xi) — a 2 b 2 x 2
Let us focus on extremal terminal state E 3 :xi(T) >0, x 2 (T) >0, y(T) >0, and T=T U At the
end of battle t = 7\ we have
(37)
w(t= 0) = aipxi(t = T) — a 2 qx 2 (t = T).
Taking (36) into consideration, we see that a point on the singular "surface"* a\b\Xi — a 2 b2X 2 yields a
positive, zero, or negative value for v(t) at t = depending upon whether q is greater than, or equal to,
or less than — . Hence, by (34) a battle trajectory which has reached the singular surface can, in general,
b 2
only remain on it for a finite interval of time ending at the end of battle (i.e., remain on the singular sur
p ii
face for 0^t^8 where 8 > 0, or equivalently T—d^t^T) when — = — . Thus, in synthesizing
q b 2
extremal trajectories we must consider three cases.
Case (a)
p_6i
q b 2
Case (b)
p b x
q b 2
Case (c)
p bi
<— .
q b 2
The solution for Cases (a) and (b) has been described by us in a previous paper [26].
If we were to plot in Figure 1 the line L' defined by a x px\ = a^qx 2 , then it would appear above, on,
or below the line L defined by a x b x xi = a 2 b 2 x 2 depending on whether q were greater then, equal to, or less
than — . This is evident from considering the slopes of these two lines.
b 2
*We refer to the locus of points such that aibiX = a 2 biX2 as the singular "surface," since if (16) holds at t = t, (hence v(t,) =0)
then a trajectory remains on this "surface" for t,^t*Ztiby use of the singular control (20). The reason for this "singular" behavior
is that, as (35) shows, — =0 so that v(t) remains identically equal to zero for a finite interval of time and hence the maximum
dt
principle fails to determine the control.
684
J. G. TAYLOR
;ase (o)
1
i 1
x'
=
* =
/j!* a 2
4>
(IA)V
; !
1lC)
X
♦*l
(IB)
■ ■ A
t =
j
J
J yS
1 >^
t
= T
**=l
— ►
t=T
(3) t=0
Figure 1. Optimal allocation for linearlaw attrition process: survivors valued in direct
proportion to their attritionrate coefficients
/ dx> \ _ a\b\ ( dxA _ a\p
aibt' \dx\) i/ azp y
since, for example,
p ^ b\ . ,. . (dxA ^ /dx 2 \
> r implies that HM > I "rM •
q o 2 \dxi/i.' \dxi/i.
The significance of the line L' and its relationship to the line L is as follows. The battle is divided
into two time phases: Phase I for ^ t ^ ti — T — T/ and Phase II for T — T/ = ti ^ t^ T. During Phase I
the optimal* target engagement policy at a point in time is determined by the location of the point on
the battle trajectory with respect to the line L, which is also the singular "surface." Above L,(f>*(t) = 0;
while below L, 4>*(t) = 1. When a battle trajectory reaches L, it remains on the singular surface through
use of the singular control </> * =
ai
During Phase II the optimal target engagement policy is to use
a.\ + a 2
<}>*(t) — 1 below L' . It may be shown that it is impossible for a battle trajectory to cross L' during
Phase II.
The above results will be developed in two sections below on the synthesis of optimal control. The
following relationships readily follow from previously developed results and are required to establish the
results of the above paragraph
(38)
v(t=0)
[>0 below L\
l<0 above V ,
"The optimality of extremals is discussed in section 7.
LANCHESTER TARGET SELECTION 685
so that
(39) r{t = T) = \ l i0TP{T) bel ° W L ''
* I forP(r) above L',
where P(* = T) = (*,(f = T), x 2 (t = 7 1 )). We also note that by (35)
f>0 below L,
(40) (r) =0onL,
l< above L.
7. DETERMINATION OF THE OPTIMAL CONTROL
As the reader is undoubtedly aware, the maximum principle only furnishes necessary conditions
of optimality. Thus, it remains to demonstrate the optimality of an extremal trajectory. Two ways in
which this may be done are as follows:
(a) show that sufficient conditions of optimality are satisfied on the extremal;
(b) by citing the appropriate existence theorem, show that an optimal control exists to the problem
at hand; there are two further subcases: (1) if the extremal is unique, then it is optimal oi
(2) if the extremal is not unique and only a finite number exist, then the optimal trajectory is
determined by considering the finite number of alternatives.
It has not been convenient to take the former approach to the problem at hand. One cannot invoke
the sufficient conditions of Funk and Gilbert [8] (which are an extension of the results of Mangasarian
[20]), since (translating into the terminology of the present paper) the right hand sides of the state
equations are not concave functions of xi, x 2 , y, and </>. Additionally, the sufficient conditions of
Jacobson [11], [12] and Jacobson and Speyer [13], [23] are not readily checked for the problem at hand.
Hence, the second approach given above is the one that we have taken in this paper. If an extremal
is unique (in the sense that only one extremal leads from the initial point in the state space to the ter
minal surface), then it is optimal and no difficulty exists. However, we have not been able to treat explic
itly the case of multiple extremals as we did in [25]. In the IsbellMarlow fire programming problem a
domain of controllability (see [25]) was determined by inequalities involving the three state variables;
in the present prescribed duration battle such a determination involves the four variables 7\, x°,
x%, yo. In other words, for the problem at hand we may consider time,*, as an additional state variable.
We have not been able to quantitatively determine the optimal control when multiple extremals exist
(although we can still give a qualitative discussion) because we could not develop time solutions for
Xi(t), x 2 (t), or y(t) (see below).
The existence of an optimal control is established by invoking an existence theorem due to Lee and
and Markus (see Corollary 2 on p. 262 of [19] (which extends some earlier results of these authors [18])).
The key aspects of being able to apply this result are the linearity of the control variable and the uniform
boundedness of responses to the controller <f>. (The latter condition is a consequence of the fact that (f>
is restricted to lie in a compact set and appears linearly in the state equations (4).) When the control
variables do not appear linearly, the existence of an optimal control is much more difficult to establish
(see, for example, [5] and [6]).
686 J. G. TAYLOR
It will be shown (see section 8) that the extremals are unique in Case (a):  = — . Hence, the
q o 2
extremal control is optimal in this case. However, the uniqueness of extremals has not been established
in the other two cases for all regions of the initial force level space. It appears as though extremals may
lead from some regions of initial force levels to both E 2 and E3 (see (6)). We have not been able to make
an explicit determination about this because it has not been possible to solve (4) to obtain a "closedform"
solution for, for example, y(t). This latter fact has precluded the analytic computation in general of the
return functional, denoted by J, in order to determine under what conditions optimal paths lead to
Ei and E 3 as was done in [25].
It would be convenient to compute the return functional corresponding to a particular control
(see [25]). Thus, we would like to express
(41) J = ry(T)px l (T)qx 2 (T),
as
(42) J = J(x° 1 ,x»,yo,T l ).
Although a state equation is readily obtained for (4) (see Table II), the author has not been able to develop
time solutions for x\{t), X2(t), and y(t) (also Pi(t), p 2 (t), and pz{t)) except for special cases. Hence,
the author has not been able to develop (42). Due to the above situation it has not been possible to obtain
an explicit expression for the switching time £1 = T — Ti. (However, this could be done for the problem
studied in [25].) Moreover, in computational studies such quantities may be numerically computed
by finite difference methods.
Finally, let us make some observations about the entries shown in Table II. When (f>(t) = for
t\ < t ^ ti, the equation for y{t ) in 1 becomes
(43) y(t) = [y'(0 + 2^i) J*gL) + 2 h {x2{t) _ Xl(tl ) }] i/,
a 2 \x 2 (ti)/ a 2
This may be obtained either by direct ingegration of (4) or by applying L' Hospital's theorem to the result
for y{t) in 1 of Table II for =£ <f> < 1. A similar expression may be obtained for y(t) when <f>{t ) = 1 for
ti =£ t =s t 2 .
8. SOLUTION SYNTHESIS WHEN SURVIVORS VALUED IN PROPORTION TO KILL
RATES
For Case (a):  = — , optimal battle trajectories are shown in Figure 1. Above the lineL with equation
q o 2
a\b\X\ = (Hb 2 x 2 the optimal control is to use <f>*(t) = until this line is encountered. When a trajectory
a 2
reaches L, the singular control <b*= ; (which keeps the trajectory on L) is used until the end of
a,\ r a 2
battle at t= T. Below L, <b*(t) = 1 is used in a similar fashion. To establish these results, we work back
wards from each possible type of end point of battle.
First, we trace extremals backwards from each extremal terminal state of battle. We give the com
plete development for E 3 : xi(T) >0,x 2 (T) >0,y(T) > 0, and T= 7\. At the end of battle t=0 equation
LANCHESTER TARGET SELECTION 687
TABLE II. State Equations for TwoonOne Combat.
1. When =£ <£(t) = constant < 1 for ti *£ t «£ t 2
4>d\
(l<t)a2
4>ai
2. fPTien < <j>(t) = constant =S 1 for ti =S t «* t 2
(10)32
X2(t)  X2(t,) (^)J
(10)32
(l4>)a 2 lAxi(ti)/ J <£a,
3. When </>(t) = &2 /or t, *£ t < t 2
ai + a 2
y(t) . [y!(t) + 2 (SLt* ) { ^±f^} M«)  *,<„»]
additionally i/aibiXi = a 2 b 2 x 2 , f/ien
y(t)= [y 2 (t 1 ) + 2a 1 b 1 (^T li ) 2 (xi(t)  x.Ct,)}] 1 ' 2
\ aia 2 /
1/2
(36) reduces to
(44) i>(t=0) = (^VoiW^r) 0262*2 (t=T)],
since we have assumed  = — . By (44) we see that there are three cases to consider depending on the
q 02
sign of the term in square brackets.
CASE(l): a,o,x I (f=7 , ) = a 2 o 2 * 2 (f = 7 , )
This corresponds to when the system ends up on the singular subarc. In this case <f>*(t= T) = a 2 l
(ai + a0, and for 0=£Ts£Ti = the "backwards time" of the first switch, we use the singular control
dv
<^*(T) = a 2 /(ai + a 2 ). Let us note that use of the singular control for O^t^Ti results in ^=° so
that
„( T ) = t ,( T =0)+ I *r =  At ti = T—Ti we switch control, since x x {t\)=x\ or x 2 (ti)=x°.
Jo dr
688 J G. TAYLOR
This yields three further subcases.
SUBCASE (1A): a,b x x\ < a 2 b 2 x? 2
At t = t x > we have that aibix" = a 2 b 2 x 2 {t^ ) < aib 2 x\ so that we cannot destroy anvmore x\.
Then we use <£* (t) = for Ti *£ r =£ T. This is consistent since v{r = Ti) =0 and
dv ,_,
r= p 3 (ai0i*i — a 2 6 2 x 2 ) < for Ti < t ^ T.
ax
(Observe that for Ti < t «£ 7\ r= a 2 x 2 y so that *2(t) > x 2 (ti).) This implies that u(t) < 0, and
ax
hence <b* (t) = for *£ t =S * x = T n.
SUBCASE (IB): a,6i*? > a 2 6 2 *°
A similar argument readily yields that </>* (0 — 1 for «= t =£ f 1.
SUBCASE (1C): a x b x x\= a 2 b 2 x\
We use (/>*(0 = a 2 /(ai + a 2 ) from the beginning.
CASE (2): ai b iXl (t = T) <a 2 b 2 x 2 (t=T)
Since v(t = 0) = ( — ] [a 161X1 — a 2 b 2 x 2 ] < 0, at the end of battle we have <f>* (t — T) = 0. Hence,
for =£ t *£ Ti = the "backwards time" of the first switch, we use </>* (t) = 0. We work backwards from
the end. Since we are above the line L, — = p 3 (a\b 1X1 — a 2 b 2 x 2 ) < 0. Hence, v{t) < Ofor all re [0, T] ,
aT
and we never do switch. Thus, we have that (b* (t) = for =£ t *£ T.
CASE (3): a 1 b i x 1 (t=T) > a 2 b 2 x 2 {t=T)
A similar argument to that used for Case (2) readily yields that <f>*(t ) = 1 for ^ t =£ T.
The above cases are shown in Figure 1. It should be noted that the above development depends
upon the fact that ps(t) > for all t. It should further be noted that, in general, trajectories (1A), (IB),
and (1C) will not all terminate in the same point as shown in Figure 1, which was drawn this way for
simplicity. Details are similar for extremals leading to E\ and to E 2 (see (6)), and are, therefore, omitted.
(The reader should recall that in these two subcases the boundary conditions for the dual variables
are given in Table I.) In this Case (a), the extremals are unique and hence optimal.
9. SOLUTION SYNTHESIS WHEN SURVIVORS NOT VALUED IN PROPORTION TO
KILL RATES
We now consider Case (b):  >t~' Again, we work backwards from each possible type of end point
q 02
of battle. We give the complete development for E 3 :xi(T) > 0, x 2 (T) > 0, y(T) > 0, and T= T\.
There are two cases to be considered.
CASE (1): Never on singular subarc for finite interval of time.
Again there are two subcases to consider, depending upon whether the system winds up above or
below L.
SUBCASE (la): a 1 b l x 1 (t=T) ^a 2 b 2 x 2 {t = T)
Since
/P2\r (pi/p 2 ) a 2 b 2 x 2  \
\ 62 /U&i/M ai6i*iJ'
LANCHESTER TARGET SELECTION 689
we see that v(t = 0) >0 and hence by (S4>) <j>*(t= T) = 1. Hence, for =£ t *S 7! = the "backwards time"
of the first switch, we use <£*(t) = 1. We work backwards from the end using this control. Since
dv
—= P3{a\biXi — a 2 b2X 2 ) >0
art
when we are below L and we stay there by using <J)*(t) = 1, we have that v(t) > for allre[0, T], and
hence we never switch. Thus, </>*(t) = 1 for < t < T.
SUBCASE (lb): a l b l x 1 (t = T) < a 2 b 2 x 2 (t = T)
Again there are two further subcases to consider, depending upon whether the system winds up
above or below L'.
SUBCASE (lbl): aib t xi(t=T) < a 2 b 2 x 2 {t=T) and a x px x {t=T) < a 2 qx 2 (t=T).
dv
In this case we wind up above L' and hence by (39)0*(f = T) = 0. Since we are above L, — < Ofor
dr
all t by (40). Combining this with (38), it readily follows that v(t) < for all re [0, T] . Thus, <b*(t) = for
s£ t =£ T.
SUBCASE (lbll): a,6,xi(*= T) < a 2 b 2 x 2 (t=T) and ai p Xl (t = T) > a 2 qx 2 {t=T).
In this case we wind up below L' at the end. By (38) and (39) we have that d(t = 0) >0 and
dv
<b*(T = 0) = 1. We work backwards from the end. Since we are above L, y < by (40) while we
ax
remain above L. Thus v(t) decreases as t increases. There are two further subcases depending upon
whether v(t) decreases to zero before the line L is encountered. Let Ti be such that v(ti) = 0. If L
has not been reached at Ti, then v(t) for t > T\ is negative and $*(t) = for t x «£ t < T. It is also
possible to just reach L when v(t\)=0. In this case (assuming that we don't remain on the singular
dv
subarc) v(t) > for t > Ti, since we pass below L and then —r > 0.
CASE (2): on singular subarc for finite interval of time.
Considering (38) and (40), it is readily seen that this can only happen when
a t biXi (t=T) < a 2 b 2 x 2 (t = T) and a t pxi (t=T) > a 2 qx 2 (t = T) .
As usual, we work backwards from the end of battle. By previous arguments it is read uly seen that
we use <£*(t) = 1 for *£ t =s n, and at t= Ti we must have a x b\Xi{T\) = a 2 b 2 x 2 {ri). We use the
singular control <J>*(t) = a 2 /(ai + a 2 ) forTi =£ t =£ t 2 . There are three further subcases.
(1) x l (r 2 )=x° l , x 2 (t 2 )<x° 2 ,
(2) Zi(t 2 )<*°i, x 2 (t 2 ) = x° 2 ,
(3) x l (r 2 )=x° l , x 2 (t 2 ) = x%
We omit the trivial discussion of these cases.
Thus we see from the above that there are six possible cases for the extremal history of combatant
force strengths in this prescribed duration battle:
(1) started below L and never reached L,
690
J. G. TAYLOR
CASE (b)£ > r
9 bg
L a, px, =a 2 qx 2
(2) u **0
(3)f**=0
Figure 2. Optimal allocation for linearlaw attrition process: survivors not valued in
direct proportion to their attritionrate coefficients
(2) always above L' ,
(3) started above L' and end up above L, but below L' without ever reaching L,
(4) end up above /., but started below L and did not remain on L for finite interval of time,
(5) started above (or on) L and were on L for finite interval of time,
(6) started below L and were on L for finite interval of time.
These six cases for extremals leading to E 3 are shown in Figure 2. Details are similar for extremals
leading to Ei and are, therefore, omitted.
It seems appropriate to make a few comments about extremals leading to E\\ X\{T) — Xi(T) = 0,
y(T) > 0, and T^S 7\. We showed in section 5 that an optimal trajectory could only reach this terminal
state (assuming that x° t > and x" > 0) by being on a singular subarc for t\ = T — T\ *£ t =£ T, and hence
P h
the dual variables have the boundary conditions shown in Table I. Thus, even in Case (b):  > — an
q 02
extremal leading to E\ lies on the singular surface for 1 1 «£ t s£ T. This situation should be contrasted with
that for extremals leading to £3 (such as (5) and (6) of Figure 2) as discussed above. In the case when
extremals may lead to both E\ and £3 from a given region of initial force levels complete details have not
been worked out (see section 7 for a further discussion). Except for this case, the extremals are unique
and hence optimal.
P 61
Case (c):  < — is similar to Case (b).
q bi
10. COMMENTS
Elsewhere [26] we have contrasted the structure of the optimal target engagement policies in
Lanchester combat when the engaged target types undergo a "linearlaw" attrition process with that
for other tactical scenarios. An important question to be answered in such studies is whether target
LANCHESTER TARGET SELECTION 691
priorities change over time. We have discovered that for the scenarios which we have so far studied
the answer to this question is determined solely by whether or not surviving target types are valued
in direct proportion to their killrate capabilities. For the case of constant attritionrate coefficients,
changes in target priorities over time can only occur when survivors are valued in excess of their kill
rate capabilities. This is true when the engaged target types are undergoing either a "linearlaw"
attrition process or a "squarelaw" one (see [27] for a discussion of the "squarelaw" case).
We now discuss how the above principle applies to the problem at hand. When a linear utility is
assigned to enemy survivors at the end of battle in direct proportion to their killrate capabilities
(as measured by their Lanchester attritionrate coefficients) against friendly forces, then the optimal
target selection policy depends only upon the location of the battle trajectory with respect to the
singular "surface" L (see Figure 1). Thus, target priorities don't change over time (they can become
equal, however). When one target type is assigned utility in excess of its effectiveness (i.e., p/q >
6i/6 2 ), then at time t, there will be a switch from tactics being determined by the location of the battle
trajectory with respect to the singular "surface" L to being determined by location with respect to
the lineL' (see Figure 2). It may be shown that t, depends on the particular battle trajectory under
consideration. Furthermore, no optimal trajectory can "penetrate" V . (The proof of this statement
is implicit in the details given in section 9. of our backwards construction technique for extremals.
If an extremal terminates at t = T above L', then <f,*(T)=0 by (39) so that by working backwards from
T we are led away from U . A similar statement holds for an extremal terminating below L'. Hence,
by construction an optimal trajectory cannot "penetrate" V .)
The methodology for solution synthesis developed in this paper is applicable to more complex
tactical situations of greater military significance. Our work here lays the foundations for the study of
the optimal allocation of supporting weapon systems (e.g., artillery, tactical air support, etc.) against
"area targets" (e.g., troop concentrations). Typical questions of interest to be answered are, "Consider
ing several infantry companies individually engaging enemy units of like size, what is the 'best' utiliza
tion of supporting artillery fires?" or, "What is the 'best' utilization of Naval fire support in amphibious
assaults?"
In a previous paper [26], we have pointed out that the structure of the optimal allocation policies
in Lanchester combat is basically determined by whether there are constant attrition returns over time
per unit of weapon system employed or diminishing returns. In the present paper we have studied
target selection with diminishing returns over time, i.e., "linearlaw" attrition process. It should be
noted that there is a problem in the literature with similar solution structures, the continuous version
of Bellman's stochastic goldmining process (see pp. 222233 of [1]). When there are diminishing
returns over time from the use of a device subject to breakdown, then the problem of maximizing the
return from use of a device in either of two potential locations has a similar structure to the optimization
problem in Lanchester combat studied here. The interested reader should compare the solution as
shown in our Figure 1 with that of Theorem 1 on p. 231 of [1] and also our Figure 2 with Figure 4 on
p. 323 of [1]. When the stochastic goldmining problem is reexamined by modern optimal control theory,
new insights are gained into the operation of maximizing the return from a resource subject to break
down or loss, and we shall discuss this in the future.
11. ACKNOWLEDGMENT
The author would like to thank the referee for his numerous suggestions for improving this paper.
In particular, the referee suggested the discussions of the state variable inequality constraints and of
the optimality of an extremal trajectory via citing an existence theorem for an optimal control.
692 J G. TAYLOR
Appendix
APPENDIX A. NONOPTIMALITY OF POLICY WHICH RESULTS IN *,(T)=0 BUT
*2<D>0
There are two cases to be considered (depending on whether or not we are on a constrained
subarc for a finite interval of time):
(1) x 1 (T) = withjti(t) >0 for TS <* < Twhere 8 >0,
(2) xi(t)=0 for t e ^t^T (t e <T).
Each of these cases requires separate treatment.
CASE (1): x t (T) = 0with*!(t) >0 for T8<t<T, where 8 >0.
There are two subcases to be considered: (1) y(T) > 0, and (2)y(T) = 0. In the first case, we have
by (33)
(A.1) v{j = Q)=qa 2 x 2 {T)<Q,
and
(A.2) ^(T = 0)=ra 2 b 2 x 2 (T)<0,
since p 2 (t = T) = p 2 (T = 0)= — q andp3(* = T) =r. Considering a Taylor series expansion ofv(r) about
t=0, one has that
(A.3) v(t)<0 forO^T<8,.
However, by (34) one sees that (A.3) implies that
(A.4) $*(t)=0 for T8i<t^T,
and hence it is impossible to have Xi(T) = but Xi{t) >0 for t < T.
In the second case in which y(T) = 0, we have from (23) thatp3(T = 0) = 0. Then, one finds that
(A.5) v(t=0) =qa 2 x 2 (T) < 0,
(A.6) J(t=0)=0,
or
and
(A.7) £(T=0)=(a 2 x 2 (T)) 2 b 2 q<0.
As above, one finds that v(t) <0 for «£ t < 82, and this again leads to a contradiction. Hence, an
optimal policy cannot result in Case (1).
LANCHESTER TARGET SELECTION
693
CASE (2): *i(0 =0 for t e ^t^T(t e <T).
Again, there are two subcases to be considered: (1) y{T) >0, and (2) y(T) =0. In the first case,
we again observe that
(A.8)
p 3 (t = T) = r>0.
It is obvious that if we have destroyed X t (i.e., X\{t e ) = for t e < T), then the optimal policy must be
to concentrate all fire on X%. Thus
(A.9)
<f>*(t)=0 {0Tt e ^t^T.
Following Bryson and Ho (see pp. 117119 of [4]) the SVIC x\ 3 s (or equivalently — *i =£ 0) may be
transformed into a control variable inequality constraint by considering the point constraint
(A.10) Xl (t e )=0,
where t e denotes the entry time to the contrained subarc, and
(All)
r = aiXiY^0 for t e < t =£ T,
at
(see also chapter 6 of [22] and [14]). The constraint «£ <)> < 1 may also be written as <j>(<f>— 1) ^0.
Then by the maximum principle [4] for (4) we are led to consider (at least formally) for t e <t^T
(A.12)
subject to:
maximize H(t, x*, pu <f>),
which by (7) is equivalent to
W~ 1) ^0,
(A.13)
maximize </)(—piai^ij+p2a2*2y) ,
subject to:
4>ai*iy=£0,
<M4>1)^0.
On the constrained subarc with xi = 0, (A.13) reduces to
(A. 14)
maximize fypza^Xzy '■,
694 J G. TAYLOR
subject to:
g( < f>) = < f>(<t>l)^0,
for all te(t e , T]. Considering the KuhnTucker theorem, we form the Lagrangian function
(A.15) L(<f>, \)=<bp 2 a 2 x 2 yk<b(<bl),
where
( = for4)*(0*l)<O,
1^0 for <*>*(<*>* 1)=0.
According to the KuhnTucker theorem, it is necessary for a maximum to (A. 14) to occur at <b*
that
dl*
(A.16) ^7 = P2a 2 * 2 yA*(24>*l)=0,
with ** ** 0 Now 4>* = ° implies that
(A.17) K* =—p 2 a 2 x 2 y^0.
(The existence of such a multiplier is guaranteed by the fact that — # at </>* = 0.) Thus, we conclude
that
(A.18) ^*=0Op 2 (()«0 for ty<t^T,
the sufficient part of the assertion (A.18) holding by virtue of the fact that (A. 14) is a concave pro
gramming problem and x 2 , y > 0.
At the entrance to the constrained subarc at t = t e , the following corner conditions hold [4], [14],
[21]:
(A.19) Pi(t;)=Pi(^)= Pi (te) fort = 2,3,
and
(A.20) H*{t) = H*{t+),
where t~ denotes a lefthand limit and H*(t~) denotes H(t~, x*, pt, <j>*). Now, we also have that
(A.21) H*(te) =p 2 (t e ){l<i>*(te)}a 2 x 2 yp 3 (t e )b 2 x 2 ,
where <f>*(t~) > and
(A.22) H* (t~) = p 2 (t e )a 2 x 2 y p 3 (t e )b 2 x 2 ,
so that the corner condition (A.20) yields that
(A.23) p 2 (t e )<f)*(t)a 2 x 2 (te)y(te) =0,
LANCHESTER TARGET SELECTION 695
with <f)*(t ) > 0. Hence
(A.24) p 2 {t e )=0.
On the constrained subarc where *,* = for t e *£ t *£ 7\ the adjoint equations read
(A.25) ^= (1 <b*)a 2 yp 2 + b 2 p 3 with p 2 (t=T)=q,
and
(A.26) ^ 3 =a 2 * 2 p 2 .
Now, (A. 18) and (A.26) yield that
(A.27) p 3 (t)^p 3 (t=T) iorti^t^T.
In the first case considered above (i.e., with y{T) > 0), (A.8) and (A.27) yield that
(A.28) p 3 (') >0 for t^t^T.
Then (A.24) and (A.25) yield that
(A.29) & (t = t e ) = b 2 p 3 (t e ) > 0.
However, if we were to have *iU) = for t e *£ t ^ T, then (A.24) and (A.29) would imply that p 2 {t) >
for t e < t < t e + y, which violates the necessary condition (A.18). Hence, when y(T) > it is never
optimal to have Xi(T) =0, but x 2 (T) > 0.
In the second case, T is unspecified and the transversality condition H*{t=T) = yields that when
y(T) =0 we have
(A.30) Pe (t=T)=0.
Now since p 2 (t— T)= — q, by (A.26) we may conclude thatp3(* — t e ) >0, and again the reasoning follow
ing (A.28) above leads to a violation of (A.18).
APPENDIX B. NONOPTIMALITY OF POLICY WHICH RESULTS IN jr,(t 2 ) = BUT
x 2 (t 2 ) > WITH t 2 <T AND x 2 (T) =
Considering the principle of optimality [1] and the result of appendix A, it is clearly not an optimal
policy to have x x {t 2 ) =0, but x 2 (h) > with 1 2 < T and x 2 (T) —0.
696 J G. TAYLOR
Let us now discuss the plausibility of the above. The result of these two appendices is intuitively
obvious when one considers marginal returns per unit of weapon system allocated. As in [26] one
considers
(£)
(B.l) = a l x u
y
Hence, as x\ is driven to zero, surviving units of X\ are increasingly more difficult to destroy. Thus, as
long as %t > 0, consideration of (B.l) shows the plausibility of the above result to the reader, since
annihilation of X\ with Xt > is accomplished under circumstances of vanishing marginal returns per
unit of weapon system allocated.
REFERENCES
[1] Bellman, R., Dynamic Programming (Princeton University Press, Princeton, 1957).
[2] Berkovitz, L., "Necessary Conditions for Optimal Strategies in a Class of Differential Games and
Control Problems," SIAM J. Control 5, 124 (1967).
[3] Berkovitz, L. and S. Dreyfus, "A Dynamic Programming Approach to the Nonparametric Problem
in the Calculus of Variations," J. Math, and Mech. 15, 83100 (1966).
[4] Bryson, A. and Y. C. Ho, Applied Optimal Control (Blaisdell Publishing Co., Waltham, Massa
chusetts, 1969).
[5] Cesari, L., "An Existence Theorem in Problems of Optimal Control, SIAM J. Controls, 722
(1965).
[6] Cesari, L., "Existence Theorems for Optimal Controls of the Mayer Type," SIAM J. Control 6,
517552 (1968).
[7] Davis, B. and D. J. Elzinga, "The Solution of an Optimal Control Problem in Financial Modelling,"
Opns. Res. 19, 14191433 (1971).
[8] Funk, J. and E. Gilbert, "Some Sufficient Conditions for Optimality in Control Problems with
State Space Constraints," SIAM J. Control 8, 498504 (1970).
[9] Isaacs, R., Differential Games (John Wiley & Sons, Inc., New York, 1965).
[10] Jacobson, D., "A New Necessary Condition of Optimality for Singular Control Problems," SIAM
J. Control 7, 578595 (1969).
[11] Jacobson, D., "Sufficient Conditions for Nonnegativity of the Second Variation in Singular and
Nonsingular Control Problems," SIAM J. Control 8, 403423 (1970).
[12] Jacobson, D., "A General Sufficiency Theorem for the Second Variation," J. Math. Anal. Appl. 34,
578589 (1971).
[13] Jacobson, D. and J. Speyer, "Necessary and Sufficient Conditions for Optimality for Singular
Control Problems: A Limit Approach," J. Math. Anal. Appl. 34, 239266 (1971).
[14] Jacobson, D., M. Lele, and J. Speyer, "New Necessary Conditions of Optimality for Control
Problems with StateVariable Inequality Constraints," J. Math. Anal. Appl. 35, 255284 (1971).
[15] Johnson, C, "Singular Solutions in Problems of Optimal Control," in Advances in Control Systems,
C. Leondes (Ed.) (Academic Press, New York, 1965), Vol. 2, pp. 209267.
[16] Johnson, C. and J. Gibson, "Singular Solutions in Problems of Optimal Control." IEEE Trans,
on Automatic Control, Vol. AC8, 415 (1963).
LANCHESTER TARGET SELECTION 697
[17] Kelley, H., R. Kopp, and H. Moyer, "Singular Extremals," in Topics in Optimization, G. Leitman
(Ed.) (Academic Press, New York, 1967), pp. 63101.
[18] Lee, E and L. Markus, "Optimal Control for Nonlinear Processes," Archive for Rational Me
chanics and Analysis 8, 3658 (1961).
[19] Lee, E. and L. Markus, Foundations of Optimal Control Theory (John Wiley & Sons, Inc., New
York, 1967).
[20] Mangasarian, O., "Sufficient Conditions for the Optimal Control of Nonlinear Systems," SIAM
J. Control 4, 139152 (1966).
[21] Mclntyre, J. and B. Paiewonsky, "On Optimal Control with Bounded State Variables," in
Advances in Control Systems, Vol. 5, C. Leondes (Ed.) (Academic Press, New York, 1967),
Vol. 5, pp. 389419.
[22] Pontryagin, L., V. Boltyanskii, R. Gamkrelidze, and E. Mishchenko, The Mathematical Theory
of Optimal Processes (Interscience Publishers, Inc., New York, 1962).
[23] Speyer, J. and D. Jacobson, "Necessary and Sufficient Conditions for Optimality for Singular
Control Problems; A Transformation Approach," J. Math. Anal. Appl. 33, 163187 (1971).
[24] Taylor, J., "Comments on a Multiplier Condition for Problems with State Variable Inequality
Constraints," IEEE Trans, on Automatic Control, Vol. AC17, 743744 (1972).
[25] Taylor, J., "On the Isbell and Marlow Fire Programming Problem," Nav. Res. Log. Quart. 19,
539556 (1972).
[26] Taylor, J., "LanchesterType Models of Warfare and Optimal Control," Nav. Res. Log. Quart.,
to appear.
[27] Taylor, J., "Target Selection in Lanchester Combat: Heterogeneous Forces and TimeDependent
AttritionRate Coefficients," Nav. Res. Log. Quart., to appear.
AN JVSTEP, 2VARIABLE SEARCH ALGORITHM FOR THE COMPONENT
PLACEMENT PROBLEM
Charles H. Heider
Center For Naval Analyses
Arlington, Virginia
ABSTRACT
The component placement problem is a specialization of the quadratic assignment prob
lem that has been extensively studied for a decade and which is of considerable practical
value. Recently, interest in component placement algorithms has risen primarily as a result
of increased activity in the field of computeraided design automation. This paper deals with
the methodology of component placement and is based on the results of considerable opera
tional experience. A tutorial presentation of tree search placement algorithms is provided,
and an improved placement procedure is described which is demonstrated to be effective in
generating near optimal solutions to the component placement problem. These solutions are
completely reproducible and are obtained at an acceptable expenditure of computational re
sources. An additional objective is an assessment of performance of the class of near optimal
algorithms. In particular, the question — how close to optimal are the near optimal solutions —
is examined.
I. INTRODUCTION
The continuing trend toward microminiaturization in electronics with large scale integration, thin
and thick film circuitry, and new wiring technology has produced radical changes in the way electronic
systems are connected, interconnected and packaged. As miniaturization increased, component and
interconnection density has become higher and higher so that the cost of designing and packaging a
digital system now has become more expensive than the components themselves. The reduction of
engineering design and packaging costs has therefore become an important objective in the goal to
minimize total system costs.
Computeraided design automation is thought by many to be the key to design cost reduction and
much has already been accomplished toward this end. Two important steps in electronic systems
design automation are component placement and interconnection routing. Performed sequentially,
a satisfactory component placement contributes greatly to the ease with which a successful completion
of the more difficult task of interconnection routing can be accomplished.*
This paper deals with the methodology of component placement. The history and relevant litera
ture of the quadratic assignment problem and its variate the component placement problem is examined
and a tutorial exposition of tree search placement algorithms is provided. An improved strategy for
implementing the Graves and Whinston [6] implicit enumeration algorithm is introduced.
The Quadratic Assignment Problem has been investigated extensively by many researchers and a
number of solution procedures have been developed.! Unfortunately, no computationally feasible algo
* Refer to Reference [12] for a survey of computeraided interconnection routing.
tRefer to Reference [14] for an extensive survey of the relevant literature and an excellent bibliography.
699
700 C. H. HEIDER
rithm has, as yet, been devised that will guarantee an optimal solution to a quadratic assignment problem
except in exceedingly trivial cases. Algorithms have been developed, however, that are capable of
generating good to nearoptimal solutions to the QAP. The fact that the available QAP algorithms
generate nonoptimal solutions is actually of little concern in many situations. The basis for this argu
ment is strictly one of economics. A nearoptimal solution obtained at a modest cost would generally be
preferred over an optimal solution obtained at a much greater expense or over no solution at all.
A second objective of this paper therefore, is to assess the performance of the entire class of
nonoptimal QAP algorithms. Using the Steinberg [15] test problem an experiment is conducted to
ascertain just how nearoptimal the solutions produced really are.
II. PROBLEM DESCRIPTION
The problem of interest can be described as follows: Given a set S= (1, . . . , m) of electronic
circuit components and a circuit board with component location set L= (1, . . . , n) n 3* m. Determine
the assignment of components to locations that will result in the minimum total length of intercon
necting wire to electrically satisfy all required circuit connections. Stated in another manner, the
problem is to determine the onetoone mapping of the set S into the set L which will result in the
minimum total wire length.
Associated with S is an n 2 interconnection matrix F= \fik\ (i, k=l, . . ., m) with/i^S 5 0. Each
fik representing the number of wires connecting component s, with component Sk Associated with L
is an n 2 distance matrix D= \dj q \ (J, 9=1, . . ., n) with dj Q 5* 0. Each dj q representing the distance
between location j and location q. The length of the interconnecting wires is determined by forming
the permuted dot product of F and D for a given mapping of S into L.
There are nl unique onetoone mappings of S into L, thus the feasible region of the problem's
solution space contains n\ point. The assignment of components to locations can be recorded by
means of an n 2 permutation matrix X = xy (i,j= 1, . . ., n) with Xy = or 1. Alternately, the mapping
can be recorded as a permutation of length n. Thus, Vi= (vi, i>2, • . ., v n ) is a permutation of the
integers (1, 2, . . ., n) or component numbers with the position in the permutation designating its
assigned location.
Using the terminology from above, the component placement problem can be stated mathematically
as follows: Given F and D, find X so as to
(1) minimize z=^ ^ ^ ^ fikd jq XijX kQ
i j k q
. (i,j,k,q=l,. . .,ra)*,
subject to
(2) 5>j=l for (i=l,. . .,n),
j
and
(3) 2*0=1 for 0=1, •'..,»)
*With no loss of generality it can be assumed that m=n. If not, then n — m imaginary components may be added with
associated /i* = 0.
COMPONENT PLACEMENT ALGORITHM 701
also letting
if component i is assigned to location j
(4) *„={J
otherwise.
In some placement problems an initial fixed cost is often incorporated. This cost is represented
by the n 2 matrix C= \cij\ (i, j=l, . . . , n) with c { j 3= 0. Here dj is the cost of assigning component i
to location; and is independent of any other component or location. This cost is exactly the cost con
sidered in the linear assignment problem. If the fixed assignment cost is to be included then the objec
tive function becomes:
(5) minimize 2 2 Ci J Xi J~^ 222 ^fikdjqXijXkq.
i j i j k q
The matrix C is frequently used to influence the final placement of the components. For example, if
component i is constrained from being placed on location j then the coefficient Cij would be set artifi
cially high so as to discourage this possibility. Equation (5) along with Equations (2), (3), and (4) represent
a more general formulation of the placement problem which has come to be known as the quadratic
assignment problem. The linear term in Equation (5) is by itself, a linear assignment problem. The
quadratic assignment problem is also closely related to the classical "traveling salesman" problem
in which the shortest travel distance through a number of cities is desired. In fact, under certain
conditions, the quadratic assignment problem reduces to the traveling salesman problem. These
conditions are:
!1 k=i+ 1, t < re
1 i = n,k=l
otherwise.
III. PLACEMENT ALGORITHMS: HISTORY AND RELEVANT LITERATURE
The problem of assigning indivisible entities to mutually exclusive locations has long been of
interest to engineers, economists, and management scientists. Beginning in 1957, Koopmans and
Beckmann [9], studying the problem of allocating plants to potential plant sites, formulated and iden
tified this problem as the quadratic assignment problem. The combinatorial nature of the quadratic
assignment problem makes the determination of the optimal solution difficult, if not impossible. Koop
mans' and Beckmann's experience led to the conclusion that the computational difficulties associated
with solving the quadratic assignment problem were insurmountable for problems of even moderate
size (say n= 10).
Complete or exhaustive enumeration is obviously impossible, as this approach would quickly
become computationally infeasible as n increased. Research was, therefore, begun to search for an
acceptable alternative to exhaustive enumeration. The literature of the quadratic assignment problem
and its related variations is concerned with the possible alternatives that have been conceived and
investigated.
Since 1957, Gilmore [5] and Lawler [10] reported optimal algorithms which were computationally
feasible only for very small problems. Gilmore reported that optimal algorithms are probably not
computationally feasible for n much larger than 15.
702 C. H. HEIDER
In 1968, the team of Nugent, Vollman, and Ruml [13] concluded, as a result of their investigations,
that the probability of obtaining a computationally feasible optimal algorithm was very remote. In
stead they suggested that interest must be focused on the development of suboptimal procedures.
Most research has, in fact, been focused on the search for an acceptable suboptimal algorithm
which would be computationally feasible for large problems and still produce acceptable solutions.
In 1961, Steinberg [15] reported the development of a suboptimal algorithm for a similar problem
which he identified as the backboard wiring problem.* Steinberg tested his algorithm on a problem
requiring the assignment of 34 components to a possible 36 locations on a (4 X 9) backboard. This
problem has subsequently become known as the Steinberg test problem and now serves as a bench
mark test for the comparison of new algorithms. Improved suboptimal algorithms have also been re
ported by Gilmore [5] in 1962, by Hillier and Conners [8] in 1966, and also by Graves and Whinston [6]
in 1966. Each of these algorithms in turn was able to better the previous best solution when tested on
the Steinberg problem. A much more recent contribution has been reported by Gaschutz and Ahrens
[3] in 1968 which has held the record for the minimum solution value to the Steinberg test problem.
It should be noted, however, that this algorithm requires excessive computation time and produces
solutions that are not reproducible on successive applications. Thus, considering computational
effort and reproducibility, the algorithm of Graves and Whinston [6] must currently be considered to
be the most acceptable to date.
Almost every conceivable approach has seemingly been investigated in the search to produce a
good placement algorithm. The more successful algorithms to date have used an implicit enumeration
scheme known as "branch and bound." The work of Gilmore [5], and Hilher and Conners [8], and
Graves and Whinston [6] have all relied on some variation of "branch and bound" techniques.
Other approaches have been the heuristic algorithms of Armour and Buffa [1] and Hillier [7].
Breuer [2] and Lawler [10] investigated the possibility of reformulation as an integer program with the
solution then being determined by integer programming techniques. Steinberg's [15] algorithm employs
a graph theoretic approach. Reiter and Sherman [16] suggested a probabilistic search procedure for
a general class of discrete optimization problems which they then applied to the traveling salesman
problem. Nugent et al. [13] investigated a similar sampling procedure which they referred to as biased
sampling. Finally, Gaschutz and Ahrens [3] produced a multistep procedure combining various ap
proaches including graph theoretic, linear programming methods, and sampling procedures.
Of all the approaches described, application experience supports the conclusion that "branch
and bound" based algorithms currently produce the most acceptable solutions when judged on the
combined criteria of solution value, reproducibility, and execution time. Reproducibility and execu
tion time can be equated in terms of dollars to the cost of attaining a particular solution value with a
given algorithm. Hence, when the expected cost of obtaining a certain solution with a specific algorithm
exceeds the expected return or value of this solution, continued use of this algorithm is no longer
justifiable.
IV. SEARCHTREE ALGORITHMS
The more common name applied to the controlled enumeration procedures to be described is
"branch and bound," the name given to the ideas employed by Little et al. [11] in their algorithm for
*As noted previously, the backboard wiring problem along with the component placement problem are variations of the
quadratic assignment problem where C = ^ the null matrix.
COMPONENT PLACEMENT ALGORITHM
703
the traveling salesman problem. "Branching" refers to the fact that in terms of a tree of alternate
potential solutions to the problem, the procedure is continuously concerned with choosing the next
branch of the tree to elaborate and evaluate. The "bound" term denotes the emphasis on, and effective
use of, means for bounding the value of the objective function at each note for eliminating dominated
paths. "Branch and bound" procedures are based on two concepts: the use of a controlled enumeration
technique for implicitly considering all feasible solutions; and the elimination from consideration of
particular solutions which are known from dominance, bounding, and feasibility considerations to be
unacceptable.
The Searchtree and the Branching Process
Originally, the basis of "branch and bound" was a twodimensional searchtree which is con
structed as illustrated in Figure 1. The vertical dimension of the tree represents the order or level of
the search and has associated with it, one of the n locations. The pairings of the locations to levels are
accomplished a priori to the application of the search procedure and remain fixed throughout the tree
search. The order in which the locations are fixed to the tree levels is critical as different pairings will
produce different permutations V g and final values Z(V g ). Gilmore suggests the following "rule of
thumb" for choosing an appropriate pairing set from the set of possible pairings. Fix the locations to
levels based on the decreasing order of
(j, q=l, . . ., n).
Start
Components
Level 1 :
Location 1
Level 2:
Location 2
Level n: j
Location n (n
FIGURE 1. Twodimensional search tree— single variable branch and bound
704 C. H. HEIDER
The free (decision) variable in the search procedure is the component to be assigned at the respec
tive levels. At level 1, the search considers the potential assignment of each of n components to the
location that has been assigned to level 1. At level 2, (n — 1) components are considered for possible
assignment to the location associated at level 2. A permutation V= (v t , v%, . . ., tfe, ■ ■ ., v n ) specifies
the assignment of the n components to the n locations, such that Vk is the number of the component
assigned to location j, at level k,j=k.* At any level k, (k — 1) components would have been assigned,
leaving (n — k+1) unassigned components to an equal number of locations.
There are a possible n\ permutations with an equal number of tree search paths and feasible solu
tions. At least one of the paths in the tree represents a minimum solution to the problem. The deter
mination of the minimum valued path is normally the objective of any search procedure. The procedure
requires (n — 1) steps to completely elaborate the tree since after the (n — 1) step there will remain
only a single unassigned component. This results in a default placement to the remaining unassigned
location.
The twodimensional searchtree has provided one of the more successful approaches reported
to date for constructing acceptable placement algorithms. Yet several drawbacks are evident which
diminish the value of this approach. They are:
1. The necessity of having to order and preassign the location to levels prior to the algorithm ap
plication.
2. The lack of reproducibility among users due to the use of different location to level assignments.
3. The probability of obtaining an inferior solution due to the order in which the locations are
considered.
All three defects are essentially related to the preassignment problem. Thus, the elimination of this
requirement should potentially improve the performance of this method.
Twovariable Branch and Bound
The next logical step in the development of tree search algorithms is the threedimensional search
tree. The addition of a second variable to the enumeration procedure extends the searchtree into
threedimensional space as illustrated in Figure 2.t Both the component and location are treated as
free variables which results in n 2 decision possibilities at level 1 versus n decision possibilities in
the singlevariable case. The threedimensional searchtree can be thought of as an inverted pyramid
with the pyramid base containing n 2 nodes.
As was the case in the singlevariable algorithm, this procedure also begins at level 1 in the search
tree and proceeds sequentially through n — 1 iterative steps. The difference is that an assignment pair
(location, component) is now chosen at each level.
The advantage of the threedimensional search procedure is that the faults attributed to the
twodimensional search are eliminated. On the other hand, the computational requirements are in
creased. This apparent disadvantage can be minimized, however, in constructing an efficient bounding
process.
This assumes that the location have been numbered to correspond to the level number to which it has been assigned.
tThe construction of an NxM searchtree was initially proposed by Graves and Whinston [6], however, it will be demon
strated that a more efficient procedure results from the construction of an NxN tree where (N — M) imaginary components
are added to the component set. The tree then has N levels as opposed to Graves' and Whinston's M level tree.
COMPONENT PLACEMENT ALGORITHM
705
Level 1
^•0
1 2
— Level 2 components
To level 3
From level n— 1
Level n
Level n location v .
/
,0"
"Level minimums
Level n component w
Figure 2. Threedimensional search tree
The Bounding Process
The method of determining the bounding or dominance measure for implicitly eliminating solutions
from further consideration is the area where tree search algorithms differ. The algorithms of Gilmore
[5], and Graves and Whinston [6] differ significantly in this respect. Before discussing these algorithms
specifically, however, the general process of bounding will be discussed, t
First, considering the twodimensional search procedure, let V g designate permutation g, and
Z{v 9 ) the criterion value of V g . Then for any other permutation Vh V g dominates Vn if Z(v g ) ^Z(vh).
Thus, in the bounding process an attempt is made to rule out of further consideration permutations
which are so dominated by another. During the enumeration procedure, the permutation V g is only
tThe use of the linear assignment problem as a bounding process, as was suggested by Lawler, has been excluded. This
is done because of the computational problems (mainly execution time requirements) necessary in setting up and solving a
large number of linear assignment problems.
706 C. H. HEIDER
partially complete (designated by V g ) since at level k, V g = (vi, V2, . . ., Vk, Ok+i, •  ., 0„). Because
Z{V g ) cannot be computed with partial V g , an estimate of Z{V g ) is therefore introduced as a substitute
in the dominance test. At level k+ 1 the bound 5* +1 is computed on the potential placement of candi
date component j from amongst the n — k unassigned components {j=\, 2, . . ., n — k) to location
k + 1 as an estimate of Z(V g ) for partial V g .
The bound Bf +1 consists of three parts, that is
B} +1 = 61 +62+63.
Letting S again be the set of unassigned components, S' the set of assigned components, L the set of
unassigned locations and L' the set of assigned locations, then B k+1 is computed as follows. First, for
the k assigned components in S' , including candidate j, the sum b\ is found through
61 = 22 /«di<oiu>* (i,jeS').
1 j
Second, the interactive contribution between members of S and members of S' is estimated from
62 = 2 YY G(fij, di(i)g) (ieS',jeS, and qeL).
Finally, an estimate of the contribution from the unassigned components is computed
b 3 = 2 2 EE G ^" dk ^ (iJeS, k, qeL).
i j k q
The available tree search algorithms differ primarily in the operator G used in the determination of
the bounds components 62 and 63. The bound/?^" 1 " 1 is computed for each candidate^' (j= 1, . . .,n—k)
with the candidate component being selected which produces the minimum value of B k+1 . The permu
tation V g (actually V g =S') is then augmented with this component and becomes V g = (vi, . . ., Vk,
Vk+i , Ofc+2 , . . . , 0„ ) . The selection process then advances to level k + 2 and repeats the bounding proc
ess. The complete enumeration of the twodimensional searchtree requires the computation of T
bounds in total where
T=2 (n+li) (i = l, . . .,n).
i
For the improved threedimensional search procedure, the selection of component and location
is again based on the computation of a lower bound, but the bound is now B k . signifying a bound with
component i fixed to location j at level A:. The minimum B k . is again selected with the minimum produc
ing pair (i,j) being made the permanent assignment at level k. The increased dimension of the search
*Read l(i) as the location of i.
COMPONENT PLACEMENT ALGORITHM 707
tree requires the computation of
bounds, compared to
for the singlevariable algorithm.
Thus, the 2variable algorithm requires
2n+l
times more computations than the singlevariable procedure. There are three bounding processes
available. They are:
(i) Minimum permuted dot product.
The operator G employed by Gilmore [5] consists of the minimum permuted dot product formed
on the proper terms of F and D. This is accomplished as follows: Given two vectors U= (ui, uz, . . .,
u m ) and W— {w\, w 2 , . . ., w m ) of nonnegative elements, the problem of determining a permutation
r of (1, . . . , m) for which the dot product
^ {UiWr(i))
is a minimum is not difficult. It is only necessary to match the smallest u, and largest Wj, the second
smallest u t and second largest Wj and so forth. For any two vectors U and W of equal dimension, the
minimum permuted dot product will be represented as P(u, w). Then the Gilmore bounds are computed
as follows.*
+ 2^2 P(/ y , d 1(i)q ) (ieS'JeS, and qeL)
i j
+ ^ P{fa,d kq ) (i,jeS,k,qeL).
i,j,k,q
In converging toward a feasible solution in successive iterations (iterations = n), the procedure of
Gilmore begins with the minimum possible (most likely infeasible) product of all elements of F and D
and attempts to maintain this value by allowing only the smallest increment to be added as a result of
making an assignment.
*In Gilmore's n* algorithm.
708 C. H. HEIDER
(ii) Value of the associated mean.
Graves and Whinston [6] used the value of the associated mean as the basis of their bounding
process which they refer to as a completion class evaluator (or CCE) rather than a bound. It has the
advantage or requiring very few multiplication operations and therefore can be computed with much
less effort than the Gilmore bounds.
Letting S, S' , L and L' be as previously defined, then the Graves and Whinston bound is computed
as follows:
B = y Jijd mi (j) (i, ;eS' )
t ' j ' ^ k f
+ {nk){nh\) (S/«) (2 *•) (I '' **• *' «*>■
The Graves and Whinston algorithm begins with the expected value of all feasible solutions and pro
ceeds to attempt to decrement this value maximally at each level in the searchtree until converging
to the final suboptimal solution value. An important feature of this completion class evaluator is that
it is easily modified to solve the quadratic assignment problem with objective function as defined by
Equation 5. In this situation, two additional components are added to the bound B which then becomes
5=61 + 62 + 63 + 64 + 65,
where
65=^3^ £ (2 Ci n ^ eS ' J eL )
i x i '
The Cij are the coefficients of the imbedded linear assignment problem.
(iii) Maximum permuted dot product.
A third bounding process not previously mentioned in the literature is suggested as a logical
extension of the previously described procedures. This is the formation of the maximum permuted
dot product in the reverse manner of forming the minimum. That is, by multiplying the largest element
of F with the largest element of D, the second largest member of F with the second largest from D
and so forth. This approach would have the effect of beginning with the worst possible value as the
lower bound and then proceeding to select the alternative at each level that minimizes the worst that
can possibly occur at each level. The value of the maximum permuted dot product when compared
COMPONENT PLACEMENT ALGORITHM 709
to the other two procedures is not immediately obvious except that the rate of change in the minimum
lower bound at each level is much greater. This aspect of the bounding processes will be discussed in
the section on convergence.
Three procedures for computing bounds have been presented with no indication of relative value
being made. Any ranking of the three methods must, however, be based in part on execution time
requirements as well as the value of the solution produced. When judged by these criteria, the mean
process is far superior with respect to execution time requirements. Only four multiplications are re
quired in the computation of each (CCE) whereas (n 2 — n)/2 multiplications are required by the other
two methods in addition to the requirement for extensive sorting of vector elements. Both multiplying
and sorting requires much more execution time than the simple additions and subtractions that are used
in the computations of the value of associated mean bound. Thus, on a computational basis, the value
of the associated mean process is far more efficient than either permuted dot product bounding process.
Additionally, the mean value process has the advantage of being applicable in solving the more general
quadratic assignment problem.
In view of this fact, the value of the associated mean bounding process is the more efficient choice
to use in conjunction with the twovariable enumeration procedure because of its considerably lower
execution time requirement.
Detailed Enumeration Procedure
The detailed enumeration procedure for the twovariable algorithm is as follows:
STEP 1. Beginning at level 1, initialize the unassigned component vector S= (1, 2, 3, . . ., n)
and the unassigned location vector£= (1, 2, 3, . . ., n). Also initialize the assigned component vector
S'= (0,0,0, . . .,0) and the assigned location vector L' = (0, 0, 0, . . ., 0). Let B* record the cur
rent minimum lower bound B§ at level k for candidate component i and candidate location j. Initially
set B* = 2 30 , an arbitrary large number. Let i be an indicator that points to the candidate component
of S that is currently being considered for permanent transfer to S' . Initially, let i = 0. Let; be an
indicator that points to the candidate location of L that is currently being considered for permanent
transfer to L' . Initially, let./ = 0.Let k be the index that records the current level of the search. Ini
tially, let k—1. Let n' designate the lengths of S' and L' and set n'—O. Let nn record the initial problem
size and set nn = n, where n is the initial length of S and L.
STEP 2. Increment i = i + l. Then remove the ith component from S and assign it temporarily
to the A:th location of S'. The length of the vector S is now reduced from n to n — 1, that is, n = n — 1.
The identification of the ith component is recorded by a marker t. Set j — 0.
STEP 3. Increment j=j+l. Then remove the ;th location from L and assign it temporarily to
the k\\\ location of L' . The identity of the ith location is recorded by a marker tl.
STEP 4. Now calculate the bound B k r by means of a bounding process, preferably the value of
the associated mean process.
STEP 5. Compare B k . with B*. If B k . «£ B*, let B* = B k ... Record the minimum bound producing
component i by i* and location j by j*.
STEP 6. Replace the >th location marked by tl back into L.
STEP 7. Test to check if all candidate locations have been evaluated, that is, if j+ 1 > n. If not,
go back to step 3.
STEP 8. Replace the ith location marked by t back into S.
710 C. H. HEIDER
STEP 9. Test to check if all candidate components have been evaluated, that is, if i+ 1 > n. If
not, go back to step 2.
STEP 10. After all candidate components in S and locations in L have been examined for possible
transfer to S' and L', select the minimum bound producing pair identified by the indicators (i*,j*).
Permanently assign i* to S' andj* to L', both to the kth position.
STEP 11. Test to check if all nn levels of the search tree have been elaborated, that is, if k + 1 >
nn. If not, set i — 0, B* — 2 30 and return to step 2.
STEP 12. Stop! The enumeration process is finished and S' and L' now contain the permanent
assignments of components and locations. Compute Z(S'(L')) to obtain the objective function value.
Convergence
There are essentially three initial bounds which can be established on the optimal solution Z*.
These bounds are easily computed from available information contained in the F and D matrices and
are:
1. Min B° — the absolute minimum lower bound on the optimal solutionZ*, most likely an infeasible
solution, which is computed from min P(f, d) with S'=0
2. Max B° — the absolute maximum upper bound on the optimal solution Z*, most likely an in
feasible solution, which is computed from max P(f, d) with S' = 0.
3. Mean B° — the expected value of all solutions, computed from
(2(/j*)2(<w)/»(»n.
M, k j,q I
The bounds min B° and max 5° establish the range of all solutions (feasible and infeasible) while
the bound mean B° separates the set of all solutions into two subsets. In all probability, Z* will belong
to the lower valued subset and will lie much closer to min B° than to mean B°, such that
min B° =S Z* < =£ mean B°«««max B°.
Convergence refers to the process through which the current active bound B k . approaches the
final feasible solution value Z(S') in the branch and bound procedure. Three convergence paths are
possible for the nstep procedure: First, by use of the min B° as the initial bound on the optimal solution,
the branch and bound procedure can be executed using the minimum permuted dot product bounding
process. At each of the nsearch tree levels, the lowest bound B k . is selected as the new bound on the
optimal solution. Finally, at level n, B?. = Z(S'), the final feasible solution value.
Unfortunately, there is no guarantee that the bound B k . will not exceed Z* at some level k, prior
to the completion of the rcstep branching process. Therefore, no statement regarding optimality can
be made concerning the application of the branch and bound procedures to the nonlinear assignment
problem. This statement also holds for the use of backtracking. Currently, the optimality of the final
solution can be established only by the following:
1. If, by chance , Z ( S ' ) = min B°, then Z(S')=Z* and is the optimal solution.
2. Otherwise, all n\ feasible solutions must be exhaustively enumerated to determine Z* and to
verify optimality.
COMPONENT PLACEMENT ALGORITHM
711
The initial rate of change in the bound B k .. will be relatively small compared with its subsequent
rate since the lower bounding rule makes the component selection at each level that produces the mini
mal incremental change in the current lower bound on Z*. However, as more and more placements are
fixed, the number of choices at each succeeding level is reduced and the incremental difference in
succeeding bounds grows larger. This effect is illustrated in Figure 3, by curve A.
it
Maximum B I —
Mean B
Minimum B
Upper bounding rule B
Mean value approximation rule C
o O
O o
°°oooo8o 7 .
ZiS'i
ooooooooooooooooo
^ — Lower bounding rule [A]
o o
1
n Level k
Figure 3. Convergence of branch and bound algorithms
The second bounding rule begins with the bound max B° and uses the maximum permuted dot
product bounding process. Here, the value max B° is the absolute upper bound on the solution Z*.
During the /istep search procedure, the bound B k . that produces the maximal decrement in the current
upper bound on Z* is selected at each level. Thus, the initial slope of this curve is much greater than
the final slope which results when fewer choices become available. The convergence curve for the
upper bounding rule process is shown in Figure 2, curve B.
A third bounding rule utilizes the value of the associated mean bounding process and begins with
mean B° as the initial bound on the optimal solution Z*. This process is essentially an approximation of
the upper bounding rule where the mean B k is substituted for max B k because its determination re
quires considerably less computational effort. The third rule is, therefore, also an upper bounding rule.
Upper bounding rules cannot be used effectively when backtracking is to be employed because domi
712 C. H. HEIDER
nance cannot be established until the final level in each path. The upper bounding rule is effective,
however, for the rastep branch and bound procedures that are of interest here for solving nonlinear
assignment problems. An example of the convergence curve for the value of the associated mean bound
ing process is shown in Figure 3, curve C. In this example, the slope will be less than for the curve B
due to the initial starting value and because of the averaging effect of the bounding process.
V. EXPERIMENTAL RESULTS
This section describes a series of experimental tests which were conducted for the purpose of
illustrating the application and the relative merits of the nstep, twovariable search algorithm. Two
test problems were analyzed; first, the simple (n — 4f) problem of Gavett and Plyter [4] and second the
larger (rc = 36) problem of Steinberg [15].
The first example is presented to demonstrate the nature of the placement problem and the me
chanics of the related solution methodologies. Since the solution of this problem is trivial, the results
obtained should not be used as the basis for comparing the performance of the various algorithms. For
the purpose of comparison, the more complex Steinberg problem is presented. The minimum solution
value for this problem is not known exactly. However, an attempt has been made to characterize the
solution space by means of sampling procedures.
The computations in the test problem analyses were performed on a CDC 3800 with the programs
written in FORTRAN. Initial work was performed on an IBM 360/75. For comparison purposes, the
CDC 3800 is approximately 45 times slower than the IBM 360/75 and roughly 2030 times slower
than the IBM 360/91 used in Graves and Whinston [6].
The Gavett and Plyter Test Problem
The sample problem which will be described was initially used by Gavett and Plyter in their
research on the optimal assignment of facilities to locations.* This problem is concerned with the opti
mal allocation of four facilities to four possible locations. Each of the facilities is required to transfer
material between the other facilities to the extent shown in the diagram of Figure 4. The interplant
material transfers can be represented by the traffic intensity matrixt F as shown in Figure 5.
The four facility locations along with their respective interlocation distances are shown in Figure 6.
Again, these values are converted to matrix form as shown in Figure 7. The objective in this problem is,
of course, to assign the facilities to locations so that the sum of the products of material quantities
transferred by distance traveled is minimized.
A twodimensional searchtree was initially constructed to demonstrate the combinatorial aspects
of the placement problem. This tree, shown in Figure 8, contains 64 nodes and 24 (nl) paths with an
equal number of feasible solutions. The solution value associated with each path was determined
through exhaustive enumeration. It can be observed in the searchtree that the optimal (minimum)
solution is 406 for the assignment (2, 4, 3, 1). The range of feasible solutions is quite broad — 406 to
607.
*The problem analyzed varies slightly from the Gavett and Plyter problem in that / 3 i = 8 rather than 5. See Figure 5. This
error was initially made by Pierce and Crowston in Reference [13], but corrected in their symmetric matrix /i*+/m. Thus, the
value produced in this study will differ slightly.
tit should be noted that the traffic intensity matrix of the facility location problem corresponds to the interconnection
matrix in the component placement problem formulation.
COMPONENT PLACEMENT ALGORITHM
713
Facilities
* Amount of
material transferred.
Figure 4. Traffic intensity diagram
c
0)
c
o
a
E
o
o
Component
2 3

10
20
5
18

9
4
8
6

8
8
15

Figure 5. Traffic intensity matrix F
The problem was solved once with each of the six possible procedures using tree search techniques.
First the twovariable algorithm was used with each of the three bounding procedures. The numerical
values of the bounds computed at each level for each of the three trials are recorded in Tables 1, 2,
and 3.
It can be observed that in each case the optimal solution was obtained, but that the order of the
location assignments was quite different. For example, the 2VBB algorithm with the min permuted
dot product bounding process assigned the locations in the order (1, 4, 2, 3) as opposed to the Gilmore
"rule of thumb" which suggested the order (2, 1, 3, 4).
The threedimensional searchtree is constructed by placing the 16 values of the "level 1" column
from Table 1 into the appropriate nodes at level 1 of the searchtree as illustrated in Figure 9. Thus,
714
C. H. HEIDER
Locations
"Unit distance.
Figure 6. Location distance diagram
Location
12 3 4
c 2
o
'&
o
o

6
7
2
6

5
6
7
5

1
2
6
1

Figure 7. The distance matrix D
it can be observed in Figure 9 that two choices existed at level 1 due to a tie for the minimum lower
bound. Either facility 1 to location 4 or facility 2 to location 1 could have been selected for permanent
assignment at level 1 based on the minimum bound value of 395. In this example, the second selection
was made. At level 2 there were then nine choices left corresponding to the nine nodes in the search
tree at level 2. The minimum bound of 395 occurred at level 2 for the assignment pair (1 , 4). At level 3
only four choices remained with the pair (4, 2) being selected on the basis of the bound 401. Finally
at level 4, the choice was made by default since only one possibility remained. The same general pro
cedure applies to Tables 2 and 3, except that the numerical values of the lower bounds are considerably
different.
COMPONENT PLACEMENT ALGORITHM
Start
Components
715
Location 1
Location 2 
Location 3
Location 4 
®®®@^(^0^S(i)S(])^^0(i)^(i)^@(l)$(i)
in 5
in
•" IP u>
>*tiaitt^it>*t<tTt in
4
Optimal
placement
FIGURE 8. Complete enumeration of n! feasible solutions
o> cm in ro
* oo o> <o
» * * *
TABLE 1. 2VBB With Minimum Permuted Dot Product Bounds
Level 1
£
a
E
'2
S
Level 2
S
s
S
"3
3
Level 3
E
a
E
"3
£
Level 4
■73
J
D.
o
/ / a
// //
°/ VV fl*
/ / J
*/Jym
/ / y
of /«•
i
1
437
i
2
467
1
2
467
i
3
419
1
3
481
i
2
4
1
395
395
*
1
4
395
*
2
2
419
2
3
403
2
4
451
3
1
416
3
2
450
3
2
465
3
2
466
3
3
3
4
406
410
3
3
3
4
406
457
3
3
406
3
3
406
*
4
1
410
4
4
2
3
406
430
4
4
2
3
406
430
4
2
401
*
4
3
470
4
4
486
4
4
522
716
C. H. HEIDER
Table 2. 2VBB With Associated Mean Bounds
Level 1
E
s
E
'E
§
Level 2
E
3
a
"E
%
Level 3
S
3
£
'E
Level 4
13
J
Q.
o
// V
oy oy
\ / /
/ ¥
oy o°/
/ y B h
oy $/
\ / /
/ >/
oy fy
/ V/ R*
i
1
513
i
2
531
i
3
495
i
4
459
*
2
2
1
2
438
495
2
1
406
*
2
2
1
2
491
479
2
3
502
2
3
444
2
3
425
2
4
525
3
1
510
3
1
466
3
1
425
3
2
524
3
2
466
3
3
3
4
496
468
3
3
444
3
3
406
3
3
406
*
4
1
484
4
1
472
4
4
2
3
463
504
4
2
415
*
4
3
489
4
4
546
Table 3
. 2VBB With Maximum Permuted Dot Product Bounds
1
1
607
1
1
503
1
2
596
1
3
585
1
3
481
1
2
2
4
1
2
508
590
533
1
4
410
*
2
1
406
*
2
1
481
2
3
607
2
3
503
2
3
425
2
4
601
2
4
494
3
1
601
3
1
490
3
1
425
3
2
607
3
3
3
4
580
523
3
3
3
4
484
503
3
3
406
3
3
406
*
4
1
570
4
2
503
*
4
3
576
4
4
607
Another set of experiments concerned the trial solution of the problem with the 1VBB algorithm
using each of the three bounding processes. Table 4 records the statistics for the application of the
minimum permuted dot product or Gilmore algorithm. The optimal solution was achieved using the
location to order pairing of (1, 2, 3, 4) as opposed to the Gilmore "rule of thumb" ordering (2, 1, 3, 4).
This same location to level ordering produced inferior solutions as demonstrated in Tables 5 and 6 with
the other bounding alternatives. The optimal solutions were obtained by using a different preassignment
COMPONENT PLACEMENT ALGORITHM
Start search
717
Level 1
Level 3
Level 4
Location 3
'Optimum solution.
Component 3 ■
Figure 9. Threedimensional search tree (GavettPlyter)
in each case. The determination of the proper order must, unfortunately, be found by "trial and error"
methods. The twodimension searchtree can be visualized by inspecting the statistics of Tables 4, 5,
and 6. The statistics of Table 5 are shown as a twodimensional searchtree in Figure 10. At level 1,
four component choices are possible for pairing with location 1 with the selection of component 2 being
made on the basis of the minimum bound 395. At level 2, three component selections are possible for
permanent assignment to location 2. Component 4 with the minimum level 2 bound 406 is chosen. At
level 3, component 3 is matched with location 3 (minimum bound 406) and at level 4, component 1 is
assigned to location 4. The final bound 406 is also the final value of the objective function Z(S'). The
final solution vector S' is then (2, 4, 3, 1).
718
C. H. HEIDER
TABLE 4. Gilmore Algorithm {1VBB)
Level 1
E
s

i
Level 2
£
9

•a
Level 3
E
9
E
'S
1
Level 4
13
J
a
O
\ / /
// ¥
// f/
/ v7 B h
yf/ b*
<?/ J/ Bl
Of J/ B fj
1
2
1
1
437
395
*
i
3
4
2
2
2
467
465
406
*
1
3
3
3
476
406
*
1
4
406
*
3
4
1
1
416
410
Table 5. 1VBB With Value of Associated Mean Bound
1
1
513
2
1
491
3
1
510
4
1
484
*
1
2
509
2
2
448
*
3
2
467
1
3
453
*
3
3
466
3
4
419
*
Table 6. 7Pflfi r^/t Max*
mum Permuted Dot Product Bounds
1
1
607
2
1
590
3
1
601
4
1
570
*
1
2
570
2
2
482
*
3
2
508
1
3
419
*
3
3
482
3
4
419
*
The Steinberg Test Problem
An experiment was conducted using the larger and more complex problem originally reported by
Steinberg [15]. This problem is reported to be an actual computer backboard layout for a UNIVAC
computer. This problem is concerned with attempting to find the optimal assignment of 34 component
modules to 36 backboard locations which minimizes the total interconnecting wire length among com
ponents. This problem is presented in order to provide a quantitative basis for comparing the various
algorithms.
COMPONENT PLACEMENT ALGORITHM
719
Start search
Level 1 : Location 1
Component
Level 2: Location 2
Component
Level 3: Location 3
Level 4: Location 4
Component
Component 
406*) Optimal solution
1
FIGURE 10. Twodimensional search tree (GavettPlyter)
The F matrix for the components is given in Figure 11. A symmetrical matrix was formed by
X (h+fji) (»,/=!, ■ . .,»)
and only the part above the diagonal was retained. This particular F matrix is sparse, that is, about
70percent zeros. The number of interconnecting wires varies from to 316 with mean equal to 15.26.
The extreme variation in F creates a large range in potential solution values. For example, the minimum
possible (infeasible) solution value is 3001 and the mean of all possible solution values is 9378.58.
Therefore, the minimum feasible solution is located between these values, that is, 3001 < minimum
Z < 9378, most likely much closer to 3001 than 9378. On the average each component is connected to
10 other components with the maximum being 26 and the minimum 1. The graph which results from
considering the components as nodes and the interconnections as links is continuous in that no disjoint
subsets exist. Thus, the problem cannot be decomposed into several independent problems of lesser
size. A total of 2625 wires connect the components.
The circuit board has 36 positions arranged to form a 4 X 9 grid as illustrated in Figure 12. The
distance matrix is constructed by determining the twodimensional Euclidean distance; that is,
d Jq = V(x r x,)H( 7j y,p.
720
C. H. HEIDER
EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
123456789 10 1112 13 14 15 16 17 18 19 20 2122 23 24 25 26 27 28 29 30 3132 33 34
Total
I
El
\ 2 17 9 4 75 7 12 22 7 1 23
170
12
E2
\y 4 16 8 16 6 4
54
6
E3
Ny 4 16 20 20 4
64
5
E4
>y 29 5 18 47 23 2 4 48 4 25
207
11
E5
\ 18 12 25 4 25 3 18 3
138
10
E6
\ 4 2 1 23 2 19 2 19
106
12
E7
\. 14 72 7 8 39 8 40 8 8 4 7 28 8
314
19
E8
N. 10 71 2 41 7 8
249
11
E9
\ 14 18
83
6
E10
\ 11 1 17 1 17 15
305
13
E11
\^316 33 8 2 8 34 6 10 6
481
16
E12
\157 25 4 1 22 1
549
11
E13
\" 6 6 583 10 9 11 2 1
486
22
E14
N. 3 11 21 1 2 5 32554
112
17
E15
\19 2 2 12 7 3
109
15
E16
\^ 6 1
34
4
E17
\ 40
40
1
E18
\^ 26
154
11
E19
\ 13 9 7 27 16 3 20 4
116
13
E20
\11 4 36 1f> 18 9 10 1 28 6 2
368
26
E21
\36 6 8 2
80
7
E22
N. 4
51
6
E23
\^ 12 9
86
7
E24
\26 5
33
3
E25
\^35 2
93
5
E26
>y 4
E1
3
G27
™i28~
74
12
Total number of wiies: 2625 \ 10 22 4 6 4 12
157
12
E29
N. 19 12
>9
8
E30
\ 19 4 5 8
99
10
E31
\^ 3 13
54
7
E32
N. 18 24
106
8
E33
\20
61
7
E34
87
8
Figure 11. Component interconnection matrix for Steinberg problem
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
Figure 12. 4X9 circuit board configuration
The distance from location 1 to location 36 in Figure 12 is for example
V3 2 + 8 2 ~ 8.54,
units.
COMPONENT PLACEMENT ALGORITHM
721
Four branch and bound trials were conducted on the Steinberg problem. First, the 1VBB algorithm
was evaluated using each of the three bounding processes. This was done in an effort to duplicate and
validate the results reported in the literature. Then the 2VBB algorithm using only the value of the
associated mean bound was considered. The reason for excluding the other two bounding processes was
based on the unacceptable large execution estimates.*
The performance statistics for each of the four trials are shown in Table 7. The published results
Table 7. Branch and Bound Algorithms Performance Statistics
Algorithm
Performance statistics
This study
Literature
Solution
Time (seconds)
Solution
Time (seconds)
Singlevariable branch and bound
A. Gilmore
4863.41
5059.38
4696.00
*88.5
*4.8
*90.0
4547.54
60
B. Mean value rule
C. Maximum permuted dot product
2variable branch and bound
D. Graves and Whinston (/VxM search tree)
4612.27
4419.49
**35.0
**37.0
4612.27
***<1.0
E. Improved algorithm (/Vx/V search tree)
*IBM 360/75
**CDC 3800
***IBM 360/91
of Gilmore could not be duplicated because the order of the location to level assignments was not
explicitly stated. The results of Graves and Whinston were duplicated by solving anA'XjW problem.
The improved 2VBB algorithm produced the best solution (4419.49) in a time much less than that
required by the Gilmore 1VBB algorithm. The final placement for the 2VBB algorithm is shown in
Figure 13.
15
16
14
29
30
33
31
3
6
1
13
12
28
32
34
22
2
18
10
7
11
20
19
21
24
17
9
8
4
5
27
23
25
26
Figure 13. Placement for 2VBB (4419.49)
* Approximately 30 min as compared to ~ 10 sec for the value of the associated mean bound.
722
C. H. HEIDER
Considerable emphasis was placed on program optimization to achieve the execution times listed
in Table 7. For example, through the use of efficient indexing schemes, the Gilmore algorithm's execu
tion time was reduced from 240 to 88 sees. This was accomplished by transforming the objective func
tion into the following:
nl n
minimize Z=^ j£ fijdi(i)i<j),
where /y is a symmetric matrix. This transformation reduces the number of multiplications and addi
tions required to evaluate the objective function from n 2 to ((n 2 — ra)/2). This indexing scheme was also
used in the bounding processes to reduce computational effort. Another optimization procedure made
use of the low density factor of the F matrix (i.e., ~30 percent). Here all zero term multiplications were
not performed in forming either the min or max permuted dot product bounds.
VI. EVALUATION OF "NEAROPTIMAL" PLACEMENT ALGORITHMS
In order to address the question of just how good are "nearoptimal" solutions, a sampling experi
ment was conducted to characterize and elucidate the solution space. Permutations of length n (36)
were drawn at random from the population of all possible feasible assignment vectors. In this experi
ment, 10,000 samples were made and evaluated. Although seemingly a large sample size, 10,000 repre
sents a mere fraction of the 3.7199 X 10 41 (36 !) possible feasible solutions to the Steinberg test problem.
The cumulative distribution curve of the 10,000 samples is shown in Figure 14. The mean of the
10
0.9
08
07
06
"x 05
04
03
0.2 
1
i — Minimum recorded
(4138.72)
(X  6. 13S)
a
• Test data
B Normal curve
n =10,000
X =9390.48
s =865.98
FIGURE 14. Cumulative distribution of 10,000 random samples
COMPONENT PLACEMENT ALGORITHM 723
sample is 5=9390.48 and the sample standard deviation* =855.98. The sample minimum was 6923.06
and maximum 12,528. The sample curve is shown superimposed over a normal cumulative distribution
function with parameters (/*, cr) equal to the sample statistics. It can be observed that the sample distri
bution appears to be nearly normal. Subsequent experiments were conducted with smaller sample sizes.
These experiments verified that a solution near fyi  3cr) could be achieved through random (stochastic)
enumeration with very few iterations (~ 1,000); however, the probability of any major improvement in
subsequent iterations is very remote.
The magnitude of this likelihood can be estimated using the assumption of a normally distributed
population.
With this assumption then, the probability that a solution, selected at random from the normally
distributed population, falls ka below the population mean is given by
1 r «**„• i/2 fxu\ •■
prob {x =£ (ft  lev)} = . I e v " '
V27TO J oo
dx.
For example, when A: = 5.0 this probability is equal to
, e v " ' dx = 0.0000002
V27TCT J oo
or approximately 1 chance in 5,000,000.
Some insight into the difficult nature of the placement problem can now be gained by considering
previously reported solutions to this problem in light of the above sampling experiment. The best
solution obtained to date is 4138.72* which is 6.13s below the sample mean. The probability of finding
this solution by random search is less than 1 chance in 1 billion. In fact, all of the branch and bound
algorithms considered were capable of producing solutions which were less than 5o below the mean.
VII. SUMMARY AND CONCLUSIONS
1. An improved placement procedure, the twovariable, nstep, tree search algorithm, was intro
duced and was demonstrated to be effective in solving the component placement problem. The salient
feature of this algorithm is the ability to consider the placement of both component and location simul
taneously. The result is that the twovariable algorithm will produce closer to optimal solutions which
are completely reproducible at a reasonable cost.
2. Searchtree techniques in general were demonstrated to be both a powerful and an efficient
means for determining near optimal solutions to the quadratic assignment problem. In the case of the
Steinberg problem, the solutions found were at least among the 0.00002percent level of the total
population of all feasible solutions based on the assumption of normality. This estimate is extremely
conservative since an infinite range on the distribution space is assumed rather than a truncated
distribution function of the actual finite solution space.
' Found by applying a pairexchange algorithm to the Caschutz and Ahrens solution in Reference [3].
724 C. H. HEIDER
REFERENCES
[1] Armour, G. C. and E. S. Buffa, "A Heuristic Algorithm and Simulative Approach to Relative Loca
tion of Facilities," Management Science Vol. 9, No. 2 (Jan. 1963), p. 294309.
[2] Breuer, M. A., "The Formulation of Some Allocation and Connection Problems as Integer Pro
grams," Nav. Res. Log. Quart. 13, 8395 (Mar. 1966).
[3] Gaschutz, G. K. and J. H. Ahrens, "Suboptimal Algorithms for the Quadratic Assignment Prob
lem," Nav. Res. Log. Quart. 15, 4962 (Mar. 1968).
[4] Gavett, J. W. and N. V. Plyter, "The Optimal Assignment of Facilities to Locations by Branch and
Bound," Operations Research, Vol. 14, No. 2 (Mar. Apr. 1966), 210232.
[5] Gilmore, P. C, "Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem,"
Journal Soc. for Industrial and Applied Mathematics Vol. 10, No. 2 (June 1962), p. 305313.
[6] Graves, G. W. and A. B. Whinston, "An Algorithm for the Quadratic Assignment Problem," Man
agement Science Vol. 17, No. 7 (Mar. 1970).
[7] Hillier, F. S., "Quantitative Tools for Plant Layout Analysis," Journal of Industrial Engineering
Vol. 14, No. 1 (Jan.Feb. 1963). p. 3340.
[8] Hillier, F. S. and M. M. Connors, "Quadratic Assignment Problem Algorithms and the Location
of Indivisible Facilities," Management Science Vol. 13, No. 1 (Sept. 1966), p. 4257.
[9] Koopmans, T. C. and M. Beckmann, "Assignment Problems and the Location of Economic Activ
ities," Econometrica Vol. 25, No. 1 (Jan. 1957), p. 5376.
[10] Lawler, E. L., "The Quadratic Assignment Problem," Management Science Vol. 9, No. 4 (July
1963), p. 586599.
[11] Little, J. D. C, K. G. Murty, D. W. Sweeney, and K. Caroline, "An Algorithm for the Traveling
Salesman Problem," Operations Research Vol. 14, No. 6 (Nov.Dec. 1963), p. 972989.
[12] Nakahara, H., "ComputerAided Interconnection Routing: General Survey of the StateoftheArt,"
Networks Vol. 2, No. 2 (1972), p. 167183.
[13] Nugent, C. E., T. E. Vollman, and J. Ruml, "An Experimental Comparison of Techniques for the
Assignment of Facilities to Locations," Operations Research Vol. 16, No. 1 (Jan.Feb. 1968),
p. 150173.
[14] Pierce, J. F. and W. B. Crowston, "TreeSearch Algorithms for Quadratic Assignment Problems,"
Nav. Res. Log. Quart. 78, 136 (Mar. 1971).
[15] Steinberg, L., "The Backboard Wiring Problem: A Placement Algorithm," Soc. for Ind. and
Applied Mathematics, Vol. 3, No. 1 (Jan. 1961), p. 3750.
PARAMETRIC LINEAR PROGRAMMING: SOME SPECIAL CASES
W. Dent
University of Iowa
R. Jagannathan
Columbia University
M. R. Rao
University of Rochester
1. INTRODUCTION
The parametric linear programming problem in which the coefficient matrix is parameterized has
been studied by several authors including Saaty [5], Courtillot [2], Willner [8] and Barnett [1]. We con
sider the general problem of the coefficient matrix being parameterized by a matrix of rank k and show
that the parametric program so defined is equivalent to a problem in which only k coefficients depend
on the parameter. Thus, for the special case of k= 1 (Willner [8]) the problem simplifies to solving a
linear program in which only one coefficient depends on this parameter (Simonnard [6]).
2. PROBLEM FORMULATION
The parametric linear program under consideration can be stated as
(1) Max c'x
subject to
(2) (A + aR)x = b
where A and R are m X n matrices;
c, x and b are respectively n X 1 , n X 1 and m X 1
column vectors;
a is a scalar parameter.
We also assume that rank (A) = m and rank (R) = A: =£ m. The main purpose of this section is to
reduce the parametric program (1) to one in which only k elements involve the parameter a. We discuss
below two procedures for accomplishing this.
METHOD 1:
We will adopt the following notation:
Ri : The tth row of R, i = 1 , . . . , m
R : The kX n submatrix of R comprising the first k rows of R.
725
726 w DENT, R. JAGANNATHAN AND M. R. RAO
/?* : The (m — k) X n submatrix of R comprising the last (to — k) rows of /?, so that R =
A : The k X n submatrix of A consisting of the first k rows of A.
A* : The (m — k) X n submatrix of A consisting of the last {m — k) rows of A so that A =
6 : The & X 1 subvector of b consisting of the first k elements of b.
b* : The (m — k) X 1 subvector of b consisting of the last (m — k) elements of b, so that b =
Suppose without loss of generality that the first k rows Ri, R 2 , . . ., Rk are linearly independent.
Then there exists an (m — k) X m matrix L such that R* = LR.
Consider the following parametric linear program:
(3) Max c'x
subject to
Ax+ ay= b
Rxy=0
(A*LA)x=(b*Lb)
*3s0; OsSa=£a
THEOREM 1: Problem (3) is equivalent to Problem (1).
PROOF: Note that Constraint (2) can be replaced by an equivalent set of constraints
Rx  y =
(4) Ax+ ay=b
(5) A*x + aLy=b*
Eliminating ay in (5) by using (4), we have the required result.
This completes the proof.
METHOD 2:
Let us adopt the following notation:
Qj
Q*
They'th column of R, j = 1, . . .,n.
The mX k submatrix of/? comprising the first k columns of/?.
The to X ( n — k) submatrix of /? comprising the last (n — k) columns of /?, so that /?
PARAMETRIC LINEAR PROGRAMMING 727
Suppose without loss of generality that the column vectors Q U Q 2 ,. . . Q k are linearly independent.
Then, there exists a A X (nk) matrix M such that Q* = QM. Therefore, R = [Q, QM] = Q[I, M\.
if
Let V= [/, M] and let V u i= 1, . . ., £, be the ith row vector of V. Then fl = J Q t V u which is
incidentally a decomposition of/? intoA: matrices, each of which is of rank one.
Consider a parametric linear program:
( 6 ) Max c'x
subject to
Vixp yi = 0, 1 = 1, . . ., k
i
x ^ 0; j3 > 1/a
where /3 = 1/a.
THEOREM 2: Problem (6) is equivalent to Problem (1).
PROOF: Constraint (2) can be replaced by
(7) [A + aXQiVi]x = b.
Setting
(8) V iX p yi =
and substituting in (7) yields
Ax + aplQ iyi =b.
Since a/3= 1, we have the required result. This completes the proof.
REMARKS:
(1) Problems (3) and (6) both have (n + k) variables and (m + k) constraints with they variables
unconstrained. Further, both problems have k parameterized coefficients (Dinkelbach [3]). In solving
the parametric linear program none of the variables y* may be considered as a candidate for exit from
the basis. Alternatively, we may replace y* by two nonnegative variables y and yr, so that^ = yt — y { .
(2) For the special case of k=l, both problems (3) and (6) have (n+1) variables, (m+1) con
straints, and only one coefficient is parameterized. The computational details discussed in [6, Section
7.10] are applicable here with the modification that the variable y is not allowed to leave the basis. Thus,
the proposed solution procedure is considerably simpler than the algorithm of Willner [8] for this
problem. See also [4, 7].
728 w DENT, R. JAGANNATHAN AND M. R. RAO
REFERENCES
[1] Barnett, S. "A Simple Class of Parametric Linear Programming Problems," Operations Research,
Vol. 16, No. 6 (Nov.Dec, 1968), pp. 11601165.
[2] Courtillot, M. "On Varying all the Parameters in a Linear Programming Problem and Sequential
Solution of a Linear Programming Problem," Operations Research, Vol. 10, No. 4 (July Aug.,
1962), pp. 471475.
[3] Dinkelbach W. Sensitivitdtsanalysen Und Parametrische Programming, SpringerVerlag, Berlin
New York, 1969.
[4] Kim, C. "Parameterizing an Activity Vector in Linear Programming," Operations Research, Vol.
19, 16321646 (1971).
[5] Saaty, T. L. "Coefficient Pertubation of a Constrained Extremum," Operations Research, Vol. 17,
No. 3 (MayJune, 1959), pp. 294302.
[6] Simonnard M. Linear Programming, Prentice Hall, Inc., Englewood Cliffs, N.J.
[7] Van de Panne C. "Parameterizing an Activity Vector in Linear Programming," Operations Research,
27,389390(1973).
[8] Willner, L. B. "On Parametric Linear Programming," SIAM J. Appl. Math., Vol. 15, No. 5 (Sept.,
1967), pp. 12531257.
SEQUENTIAL SEARCH OF AN OPTIMAL DOSAGE:
NONBAYESIAN METHODS*
B. H. Eichhorn
Case Western Reserve University
1. INTRODUCTION
The present research represents a class of onesided (sequential) stochastic approximation pro
cedures, which have possible applications in many fields. A motivational background could be simply
described in terms of a biomedical problem. A drug could be administered at various dosages. High
dosages although desirable increase the level of toxicity. The dosagetoxicity relationship is not com
pletely known. We want to use sequential experimentation to approach from below the highest possible
dosage one may give without causing too much toxicity. For convenience let us use this biomedical
terminology.
Let x designate an assigned dosage. This variable can be controlled by the experimenter. Let
Y(x) designate an observable random variable, which is the toxicity level associated with the dosage x.
The distribution function of Y(x) depends on x in a manner specified later under the various statistical
models. Generally, the expected toxicity is an increasing function of the dosage. A threshold of toxicity
is specified in the sense that higher toxicity levels are undesirable. For the sake of simplicity, assume
that the toxicity threshold assumes the value zero. The objective is to assign the largest possible dosages
without crossing the threshold of toxicity. Since toxicity levels are not determined completely by the
dosages, a tolerance probability y, < y < 1, is specified, and the optimal dosage, £y, is defined as
the largest x value under which the probability that Y(x) < is not smaller than y; i.e.,
(LI) P[Y(iy) =s 0] 2* y.
If £y were known it would have been used in each case. In certain models one could make simple
transformations and consider instead of £y the value of x for which E{Y(x)} = 0. This value is denoted
by £ . In all the models treated here both £ and ijy exist and are unique. In the remainder of the present
section we use £ to designate the optimal dosage in either one of the above cases.
A sequential search procedure is a procedure of determining a sequence of dosages X\, x 2 , . . .;
where for every n > 1 x n is a measurable function of (x\, . . ., x n i) and of (Y(xi), . . ., Y(x n i)).
We consider procedures which satisfy the following two conditions:
(i) Feasibility— For each n, n—1, 2, . . ., and a preassigned sequence {<*,,; n&l}, where
0<a„< 1,
(1.2) P[xn>Z]*Za H .
* Partially supported by Project NR 042 276 of the Office of Naval Research at Case Western Reserve University.
729
730 B. H. EICHHORN
(ii) Consistency— x n converges in probability to £. If almost sure convergence holds we speak of
strong consistency.
The value of a„ represents a bound on the probability of exceeding the optimal dosage £ at the nth
stage. The values of ot n can be held fixed, or can be decreased as information on £ increases. In special
cases <x n can be chosen so that V oc„ =£ e, without losing consistency. In such cases the overall prob
n=l
ability of exceeding £ anywhere in the sequence is smaller than e. We further notice that at the nth
stage, the probability that the observed toxicity Y(x„) will exceed the threshold is bounded by (1 — oc n )
(1 — y) +a n . In the present paper we restrict attention to the case of fixed a.
There are many feasible and consistent sequential search procedures for any one of the considered
problems, and it is desirable to introduce some optimality criteria. Due to the complexity of the prob
lem we do not set rigid optimality conditions. However, some of the procedures that will be presented
have certain optimal properties, which will be discussed in the sequel. In a recent paper [1] the search
procedure was studied for the special cases of linear regression E{Y(x)} = a + bx, with a conditional
normal distribution of Y(x) at each x. In that paper the intercept and variance were assumed to be
known. NonBayesian and Bayesian procedures were develped. The present paper provides solutions
for cases in which either the slope or the intercept is known and the variance may be know or unknown.
Only nonBayes procedures are discussed here. In section 2 the statistical model is explicitly specified.
The case of unknown intercept is discussed in section 3. Section 4 is devoted to the case of unknown
variance. The case of both intercept and variance unknown is discussed in section 5. Section 6 treats
the case of both slope and variance unknown. The general case of linear regression in which all the
parameters are unknown is subject for further research.
2. THE STATISTICAL MODEL
The following assumptions are kept throughout this paper. Let Y(x) designate a random variable
representing the observed response at dosage x. We assume that the conditional distribution of Y(x),
given x, is normal with mean h(x) and variance (r 2 (x). We further assume that h{x) is a linear function,
h(x) = a + b x, with a < and b>0 over the interval of interest =S x *£ k. The optimal dosage £
(or £y) belongs to this interval. With respect to the variance <r 2 (x) we distinguish between two models:
Model I; <r 2 (x) = x 2 cr 2 , and
Model II; <r 2 (x) ~(J 2 ; where cr 2 is a positive constant.
For the sake of simplicity we assume that the maximum allowable toxicity is f) = 0.
A general formula for the optimal dosage £y is given by:
/ 2 i\ t _ \~ a/b + zycr), in Model I
I— (a + zy<r)lb, in Model II;
where zy is the yfractile of the standard normal distribution. In cases of Model II, with known <r we
can further simplify by making the transformation 17— ► tj — zyo again taking it to be zero. In this case
£y assumes the simple expression —alb, which will be denoted by £.
3. CASES OF UNKNOWN INTERCEPTS
We assume here that the slope b and variance cr 2 are known.
SEQUENTIAL SEARCH 731
3.1. Model I
We start the study of cases with unknown intercepts with Model I. Here we have to assume the
knowledge of a value x* such that 0< ** *£ £ y . A search procedure is defined as a sequence X u
^2, ... of dosages each being a function of all former dosages and observations of toxicity. For
Model I we define the search procedure P.l. as follows:
Define:
Y t =Y(Xi), £=1,2,. . .
U i =Y i bX i , £=1,2,...
(3.1) U n =f t U i /n, n=l,2, ...
i=l
and
£n = (U„ + <rkl\/na)l(b + zy<r), n = 2, 3, . . .
ii =  (tfi+<r**/Va)/(& + zytr).
The initial dosage is taken to be
X, = x*,
and successively we set
(3.2) X n+ i = Max (**,»), n = 2,3, ....
THEOREM 1: The search procedure P.l. is feasible and consistent
PROOF: The random variables Uu £=1,2,. . . have conditional normal distributions, given
Xi, with mean a and variances x 2 <r 2 .
These variables are uncorrected, but not independent. Indeed for every i < j :
(3.3) EiUiUj} = EWiEWjl&i]} = E{U ia )=a\
where & ' \ denotes the (rfield generated by the first i random variables. Thus, E{0„} = a and the
variance of U n is bounded by cr z k 2 ln. Hence, from the Chebychev inequality we infer that £„is a 1—a
lower confidence limit for £y = T~r; • Furthermore, by assumption** *£ gy. We therefore conclude
that the procedure P.l. is feasible, i.e.,
P(X n +i *S £y) > 1  « for each n 5= 0.
The consistency of P.l. is established from the fact that Var (U n ) ^ cr 2 k 2 ln. Hence U n converges
in probability to a, and akj Vna — > as n — >°°, • Hence „ * £ in probability. (Q.E.D.)
732 B H EICHHORN
We remark that the procedure P. I. is based on a rather crude inequality, and the upper limit k
might be very large. Thus, although the procedure has the required properties it may be inefficient
for small values of re. The efficiency of the procedure could be improved by further investigating and
employing the properties of the distribution of U n 
3.2. Model II
Whenever Model II can be assumed, i.e., cr 2 (x) = cr 2 for all x, we can attain stronger results. Let
Ui • i= 1, 2, . . . and U„n = 1,2,. . . be defined as in Model I, and set
(3.4) in = Unlb<rz l  a lbVli, re=l,2
The sequence of dosages under search procedure P*.l. is determined in the following manner:
The first dosage is *i = 0, and for every re = 2, 3, . . . the dosages, x n , are specified by
(3.5) x„=Max(, f ,_ 1 )
THEOREM 2: The search procedure P*.l. is feasible and strongly consistent.
PROOF: Ui, (i=l, 2, . . . , re) are independent identically distributed (i.i.d.) random variables.
They have a normal distribution with mean a and variance cr 2 . Since a is negative £ is positive, and
Xi < £. Furthermore, U n is normally distributed like N(a, cr 2 jn). Hence f n is a 1 — a lower confidence
— a *
limit for£ = — j—. Moreover, £ n is a uniformly most accurate lower confidence limit (U.M.A.; see Lehmann
[2], pp. 7881). This property will be further discussed later. The strong consistency oi P*.l. is shown
— a a r' * — I — fit S
as follows: We have Un 1 * a, and (TZiJ V/i— »0 as n— *■ °°. Therefore £« = — £/„/6 — oZi  a /ft Vre — »
alb = £\ and*„ + , = Max(0,£,,)^'£>0. (Q.E.D.)
The procedure P*.\. has a certain optimality property that will be discussed now. Define the
dosage "shortage" at the nth trial as R n = (£—X n ) + , which is the positive part of the distance from^„
to the optimal dosage £.
A procedure will be called optimal if it is feasible and for each N, N=l,2, . . .it minimizes the
expectation of the total shortage in the first N steps, i.e.,
~>
(£«<)
THEOREM 3: The procedure P*.l. is optimal.
PROOF: We shall show that for each n, n=l, 2, . . ., the procedure P*.l. minimizes the ex
pected shortage R n , with respect to all feasible procedures. Since this property holds at each stage
independently of what happens at all other stages the optimality of P*.l. will follow.
Consider the nth trial, and suppose that an arbitrary feasible procedure n assigns it a dosage
x n  x n is a function of the first (re — 1) observations only. Since II is feasible, x n is a lower confidence
limit for £ at level (1 — a). Hence, the UMA property of£„_i implies that P f {£„_i *££'} ^P({x„^^'}
for all < £ < oo, and all f < £ . We notice that if £ > then x„ ^ £' if and only if /*_, *£ £'. On the
other hand, if £' < 0, P({x„ ^ £'} = 0. Therefore,
(3.6) P((x n ^i')^P((x„^i'), foranyO^<°o, and £' < £.
SEQUENTIAL SEARCH 733
Let 8 = ££'.
It follows from (3.6) that
(3.7) Pd^x„^8] ^P([£Xn^8], for all 8 > 0.
In analogy to R n , define /?„=(£ — x„) + . Inequality (3.7) is equivalent to
(3.8) P ( [R n ^8]^P ( [R n ^8], for all 8 > 0.
Since both R n and R n are nonnegative random variables we infer that for every n—\, 2, . . . .
(3.9) Ef[Rn]^ EdRnl
(Q.E.D.)
4. SEARCH PROCEDURES FOR CASES WITH UNKNOWN VARIANCE
In the present section we consider cases in which a and 6 are known and cr 2 is unknown. We
assume that cr «£ cr, where cr is a given constant. Although x = does not mean here that we do not
administer any drug, there is an "absolute zero" dosage level under which we cannot go. Therefore
cr has to be taken in such a way, that if cr= 5" we shall still get £ larger than this absolute zero dosage.
It will be simpler to allow here also negative values for x. In these cases we naturally search for £
as given by (2.1).
4.1. Model II
We start with Model II for which £ as in (2.1) assumes the following form:
(4.1) £. y =£Ccr,
where £ = — a/6 and C = Zy/b. Both £ and C are known.
The search procedure which is proposed for this case, and which is designated by P*.2. is speci
fied in the following manner:
Let xi =  — Ccr. For each i=l, 2, . . . let Ui=Yi — bxi — a, and
Furthermore, let <r* >a = Min (S„lx 2 „, a , cr 2 ), where x„, a is the ath fractile of the chisquare distribution
with n degrees of freedom.
The dosage for the (n+ l)st trial is specified by P*.2. as
(4.2) *n+i = £ — Ca„, a , n=l,2, ....
THEOREM 4: The procedure P*.2. is feasible, strongly consistent and optimal
734 B  H EICHHORN
PROOF: Ui(i=l, 2, . . ., n) are i.i.d. normal random variables with mean and variance cr 2 .
Therefore S 2 /cr 2 has a chisquare distribution with n degrees of freedom. Hence
(4.3) P[S 2 Jcr 2 ^x 2 n, a ]=P\.cr 2 ^S n lx 2 n , a ] = a,
and cr 2 a is a U.M.A. upper confidence limit for cr 2 , at level 1 — a. This implies the feasibility of the
procedure. To show strong consistency it is enough to show that cr 2 ua N ^£±+ cr 2 .
Let us write
S, 2 /x,i a = (lf/ 2 /«)wx 2 n> J.
According to the Strong Law of Large Numbers and the Central Limit Theorem we conclude that:
n
V Uf/n^ cr 2 as n—> °°, \l, a^ n + Z a V2n as n— * °°.
i=l
Hence, Xn,al n ~* L As a ^ so °~ 2 < ^" 2 we obtain a 2 a ^i^a 2 
As cr 2 „ is a U.M.A. upper confidence limit for cr 2 , at level 1 —a, it follows that x n+ i = ^ — Ccr n ,a is
also a U.M.A. lower confidence limit for gy, (fjy = t; — Ccr) at level 1 — a, for n= 1, 2, . . . . The opti
mality of P*2. follows then, by the same arguments as in the proof of Theorem 3. (Q.E.D.)
4.2. Model I
A search procedure having similar properties can be found for Model I just by a few modifications.
Thus for Model I we set
Ui=(Yibxia)/xi i=l,2, ... .
Ui(i=l, 2, . . ., n) are again i.i.d. normal random variables with mean and variance cr 2 .
According to (2.1) £ y =_ a/(6 + z y cr). We employ also here the statistic cr 2 a = S 2 /x 2 , a which is a
U.M.A. upper confidence limit at a level (1 —a). We thus define the search procedure P. 2. for Model
I by specifying the dosages:
X\ = — a/(b + zycr) ,
(4.4)
x n+ i = al(b\zycr* a ), n=l,2, . . .,
in which cr* a = Min (cr n>a , cr).
THEOREM 5: The procedure P.2. given by (4.4 ) is feasible, strongly consistent and optimal.
PROOF: By the previous arguments, cr 2 a s^. cr 2 and x„ 2^» £ as n— »°°. Again the fact that
cr** a is a U.M.A. upper confidence limit at level 1 —a for cr 2 , implies that x„+i = — a/(b + zycr* a ) is a
SEQUENTIAL SEARCH 735
U.M.A. lower confidence limit for £ = — a/ (6 + z cr), at level 1—a. This implies the optimality of the
search procedure P.2. in the same manner as for procedure P*.2. and P*.l.
5. CASES WITH UNKNOWN INTERCEPT AND VARIANCE
We assume here that the slope b is known, but the intercept a and the variance cr 2 are unknown.
We have a solution only for Model II, where cr 2 is fixed. The search is for £ y which is specified in (2.1),
and is subjected to the further condition that £y > 0. Since here we need at least two observations in
order to estimate cr, take the first two dosages at ;ti = ;c2 = 0. Let Ui—Yj — bxi i—l, 2, . . .). Also
here the Ufa are i.i.d. random variables having a normal distribution N(a, cr 2 ). Let
U n = Y,U t ln
and
S 2 »=i (U t du)*l(nl)
for n — 2, 3
Notice that the expectation of btj +U n is bg y + a = — Zycr. Hence Vn (bij y + U n )IS n has a non
central t distribution with n — 1 degrees of freedom and noncentrality parameter — vnZ y . Let t a [n — 1 ,
— vnZ ] denote the afractile of this noncentral t distribution. Then,
(5.1) Pa,*[V^(b£ Y +U n )ISn^t a [nl,V^Z y ]] = la, for all a, cr.
Thus,
(5.2) £n, a = S„t a [nl, ViiZ y ]lV^bU„lb
is a U.M.A. invariant l = a lower confidence limit for (j y If A"„ + i = Max (0, „, a ) for n = 2, 3, . . .we
have a feasible procedure.
To prove strong consistency we have to show that  ^^ f , This follows immediately from the
fact that U n ^ a and S» s^ o, and that (t„[nl,  V^Z y ]/ V^  Z a / V^ + Z v ) »0 as n > ».
Indeed < a [n — 1, — Vra Z y ] is asymptotically like the ath fractile of the normal distribution with mean
— ZyVn (the noncentrality parameter) and variance 1. Define now the sequential search procedure
P*.3. Accordingly let *i = * 2 = and *„+i = Max (0, £„, „) for all n = 2, 3, ....
The same arguments as used in the proof of Theorem 3 imply that the procedure P*.3. minimizes
x
the value Y E(£;y — Xi) + for every fixed integer N, among all feasible procedures, which are invariant
under linear transformations on the values of Y t . We shall call this property invariant optimal. We
have then proven:
736 B. H. EICHHORN
THEOREM 6: The sequential procedure P*.3. is feasible, strongly consistent, and invariant
optimal.
6. CASES OF UNKNOWN SLOPE AND VARIANCE
In the present section we provide a solution only for Model I. Here we define the auxiliary variables
U\ , U2 , • • in the form
(6.1) Ui=(Yia)lxi, i = l, ... .
Then £7,, t = l, 2, . . . are i.i.d. random variables normally distributed with expectation b and
variance cr 2 . U„ and S 2 are as before the sample mean and the sample variance based on the first n
l7,'s. Routine arguments concerning the distributions of these statistics yield immediately that
Vn(U„^ y + a)l^ySn has a noncentral t distribution with n — 1 degrees of freedom and noncentrality
parameters — Zy V^.
In order to obtain a feasible procedure we assume the knowledge of a value x* such that 0< x*<£y.
Define the procedure PA. in the following manner:
Let xi = x 2 = x*, and
(6.2) z„+i = Max (x*,£ n «), n = 2,3, . . .
where
(6.3) gn,a=al(U n =S n t a [nl,Zy V7t]\V7l).
THEOREM 7: The sequential procedure P.4. is feasible, strongly consistent, and invariant
optimal.
PROOF: From the definition of t a [n — 1, — Zy Vn ] we obtain that
(6.4) P b ,o{{y^alU„S n t a [nl,ZyVn~]lVn^)} = la.
Hence P{£y 3 s 1jn,a} = 1 — a. This proves feasibility. For consistency we notice that U n ~* b a.s.,
S n *■ cr a.s. and t a [n — 1 , — Zy Vn ] / V~ rr> — Zy as n »• 00. Hence, according to (6.2)(6.3) x n * ijy a.s.
Finally, the invariant optimafity follows exactly in the same way as for P*.3. (Q.E.D.)
REFERENCES
[1] Eichhorn, B. H. and S. Zacks, "Sequential Search Of An Optimal Dosage I," submitted for publica
tion to the Journal of American Statistical Association (1972).
[2] Lehmann, E., Testing Statistical Hypotheses (John Wiley & Sons, New York, 1959).
FURTHER LIGHT ON NONPARAMETRIC SELECTION EFFICIENCY
Edward J. Dudewicz*
The Ohio State University
and
Chunglien Fan
The University of Rochester
ABSTRACT
A nonparametric selection procedure &> B s was proposed by Bechhofer and Sobel (1958)
and studied by Dudewicz (1971) in comparison with other procedures under normal and
uniform alternatives. He found 0> BS always required larger sample sizes, sometimes sub
stantially so. For 2point populations we find more extreme results. We also find that 3P B s
may be substantially better than reasonable competitors designed specifically for 2point
populations. Finally, a new nonparametric selection procedure (conjectured to be better than
SPbs) is proposed.
1. INTRODUCTION
Let tt\ , 7T2, . . . , 77> be A: (3= 2) populations such that if an observation X is drawn from 7r, then
(1.1) P[X=c i ] = lp,P[X = s + c i ]=p (i=l k),
where 5 (5 > 0) and p (0 < p < 1) are known and c x , . . . , Ck are unknown. We assume that the associa
tion between tt\, . . ., TTk and qi], . . ., q*] (where qi] ^ • • • =* q*j denote the ordered C\, . . .,
Ck) is completely unknown, and that the best population is that with the largest location parameter q*j.
In this paper we consider the problem of selecting the population associated with q*] = max (ci,
. . . , Ck). The procedures we will consider each take n independent observations per population in a
single stage, where n is set so that the probability of a correct selection (CS) satisfies
(1.2) P(CS) 2= P* whenever c [k]  q fc _,] 3* 8*
where P*(l/k < P* < 1) and 8*(S* > 0) are specified in advance by the experimenter.
In Section 2 we consider a procedure based on sample means, say^V Let Xi denote the sample
mean of the n observations from 77, (1 *£ i ^ k) , and denote the ordered Xi , . . . , X k by X{ X ] =S . . .
^X[k). &M selects that population which yielded X[ k ] (the largest sample mean). Note that (fori 4=;')
P[Xi — Xj] = unless for some =£ /, m =£ n we have
ncj + Is = ncj + ms
*This author's research was supported by ONR contract N0O014^68AO091 and by the U.S. Army Research Office
Durham.
737
738
(1.3)
E. J. DUDEWICZ AND C. FAN
(/ — m)s = n(cj — ct)
I — m = n(cj — ci)/s.
Since /m is an integer, (1.3) holds iff s \ n (cj — c* ) , which is false for almost all real s.
In Section 3 we consider a procedure specifically designed for twopoint populations, say £P S  This
procedure is intuitively appealing (and, for twopoint populations, may be optimal among all single
stage procedures which take n observations from each population). Some applications, the most
important of which deals with the problem of the repair (or replacement) of the worst of a group of
machines, are given; however, the main values of our study of ps are probably the light shed on the
efficiency of a certain nonparametric selection procedure (see Section 4) and the expository value.
In Section 4 we use&M and SPs to shed further light on the efficiency of a nonparametric selection
procedure suggested by Bechhofer and Sobel (1958) and studied by Dudewicz (1971).
In Section 5 we propose a new nonparametric selection procedure, which we hope to study in
detail in a later paper.
2. THE MEANS PROCEDURE
Recall that we take n independent observations per population in a single stage. For the means
procedure 8?m (see Section 1), the P(CS) is minimized (over q*] — q*i] 3 s 8*) when C[i] = . . . = C[ki]
= C[k] — 8*, and is then
(2.1)
P L (CS) = J (f(j + n *1 )) *~p(q fc] + J  ,) .
where
(2.2)
p(q*] + ^sW n W(lp)"J = 0,1 n),
and where
(2.3)
F(x)={
( 1
[x]
if x > n
JfjQp'dp)' ifO^x^n
if x < 0.
i=0
I
Note that P L (CS) = 1 if 8* > s (iff (say) \* = 8*/s > 1), in which case (1.2) can be trivially satisfied by
taking n= 1. Hence we now assume 0< 8* < s (i.e. 0< X* < 1). (As a check of (2.1) note that Pl(CS)
= lifp = 0orp= 1.)
If p were unknown, one might desire to set n so that
n /[J+"^*J/n\ \ fcl/ n \
(2.4) inf P L (CS)= inf f( V ( P'd  p)""') ( Ml p) »■> = />*.
NONPARAMETRIC SELECTION 739
The p at which Pl(CS) is minimized is a function of n, k, A* for which we have no general simple ex
pression. When n = 1 3?m is the same as's, and an exact solution has been obtained in Section 3. When
A: = 2 it is easily shown directly that Pl(CS) is minimized at p = 0.5. (Calculations of Pl(CS) for p
= 0.00(0.01)1.00 at {n= 1(1)5; k = 2, 5; A* = 0.1 + (0.1 + )0.9+} show that the minimizing p is between
0.33 and 0.50, and that P L (CS) at the inf differs little from P L (CS) atp = 0.5. )
An interesting phenomenon showed up in our claculations: Pl(CS) at fixed p, k,K* does not neces
sarily increase as n increases for small n. (E.g. for k = 2, p = 0.5, A* = 0.1, (ra, Pl(CS) ) takes on the
values (1, 0.75), (2, 0.69), (3, 0.66), (4, 0.64), (5, 0.62). Note that n\* is not an integer in any of
these cases; as we say at (1.3), this is necessary in order that ties have probability zero.)
3. THE TWOPOINT PROCEDURE
We now wish to specify the twopoint procedure ^s precisely. First we take n independent observa
tions per population in a single stage. These allow as to conceive of decomposing
(3.1) n={77i, . . .,TT k }
into three disjoint (random) sets with union II , namely*
_ [ir.TTeU. and the sample from tt yielded!
1 its lower value n times J
[tr.TreYl and the sample from tt yieldedl
(3.2b) °ll = \ . . .
[its upper value n times J
_ f7r:7T€n and the sample from tt yielded both its upper)
1 and lower values at least once each J
We then select that tt which yielded the single** largest value.
We now wish to find the P(CS) for procedure's Let Tr (k) denote the best population, and let v+ 1
denote the number of c< + s which exceed q*]. Then
P(CS) = P(CS  TT (k) e^)P(TT (k) e^)+P(CS \ Tt (k ^)P(TT (k )^) + P(CS \ TT {k) eF)P(TT {k) er)
(3.3) =((lp)»)"(lp)"+l p n +l • (lp»(lp) n )
= l(lp)"+ ((lp)")" +1 .
*In practice we cannot distinguish among members of Jf 01 ^.
"Reasoning as at (1.3), ties can occur if and only if:
(1) q*ij + s = q*j + s, or (2) q*_i] + s = c [k] , or (3) more than two c ( + s exceed C[»] and at least two of these are equal.
Since if q*  1 j + 5 = q*j + s we have two "best" populations, it is easy but tedious to verify that (even if ties occur and we randomize
over tied populations) equation (3.3) is still valid in cases (1) and (2), while in case (3) the P(CS) is increased slightly. We exclude
case (3), considering such configurations only as limits (in which case we consider the limits of probabilities as this configuration
is approached, and not the probabilities in the limiting configuration).
740
E. J. DUDEWICZ AND C. FAN
This P(CS) is minimized (over C[k] — C[fci] 3 s 8*) when C[i] — •
maximized to v = k — 1) and is then (assuming < 8* < s)
= C[ki] = C[k] — 8* (i.e. when v is
(3.4)
/MCS) = l(lp)»+(lp)«*
In many cases one will not know p, and may desire to set n so that
(3.5)
inf P,.(CS)=P*
osspsn
The Pl(CS) is minimized at p = 1 — It) . (The following table illustrates typical values.) Note that
at the minimizing p the probability is not dependent upon n and (3.5) cannot in general be achieved
when p is unknown:
(3.6)
inf Pl(CS) = 1 (!/*)*» + (l/yt)*" 1 .
Ospsl
1
i
(l/*)** 1
i)n
inf P L (CS)
\ n
1
2
3
4
5
6
7
8
9
10
Any
k \
2
.50
.29
.21
.16
.13
.11
.09
.08
.07
.07
.75
3
.42
.24
.17
.13
.10
.09
.08
.07
.06
.05
.62
4
.37
.21
.14
.11
.09
.07
.06
.06
.05
.05
.53
5
.33
.18
.13
.10
.08
.06
.06
.05
.04
.04
.47
6
.30
.16
.11
.09
.07
.06
.05
.04
.04
.04
.42
7
.28
.15
.10
.08
.06
.05
.05
.04
.04
.03
.38
8
.26
.14
.09
.07
.06
.05
.04
.04
.03
.03
.35
9
.24
.13
.09
.07
.05
.04
.04
.03
.03
.03
.32
10
.23
.12
.08
.06
.05
.04
.04
.03
.03
.03
.30
Remark. Note that if 7T(*)€ ff~ u °U we always make a correct selection. However when 7T(fc)€oSfwe
won't make a correct selection unless all populations tti such that c* + s > q*] are also in=Sf. One might
try to modify &$ in order to increase the chances of a correct selection when TT(k)iJ^, reasoning as
follows. Any of three configurations of "best in^" vs. "best m^^W can occur:
"Best in ^ r "
"Best in if U <&"
i i
a
a+s
a
a+s
a a+s
b
b
b
Case
(i)
(ii)
(Hi)
NONPARAMETRIC SELECTION 74^
se
In situation (i) ((in)) we select the population yielding (a, a + s) ((b)), and are certain of having
lected correctly. In situation (ii) ir (fc > could be either in 5" or in Jf u <%.0> s always selects the best in^ 1
this case. One might also study rules which select the population yielding (b) in case (ii) (e.g. so select
when p < Co). We have not chosen to do so since (1) such a rule does not uniformly increase the chances
of a correct selection, (2) such a rule is not as simple and intuitively appealing, and (3) such a rule is
dependent upon knowledge of p (whereas SP S can be utilized regardless of the value of p . . . although
of course its characteristics depend upon the value of p).
Applications
Suppose we are considering k machines with the following characteristics. The output of each
machine is either "a" or "a + 5" with respective probabilities of "1 — p" and "p" (two modes of opera
tion). The probability "p" may be known or unknown, but is the same for all machines (same modal
probabilities). However, the value of "a" may be different for different machines. We may wish to select,
after observing some outputs, that machine with the largest "a" (the one that has "slipped" the most)
and repair or replace it (repair or replacement of the worst of a group of machines).
As a second application (this one hypothetical) consider mining. Suppose that a lode (location
unknown) results in precious stones being scattered at sites in an area of 10 (say) square miles. Each
site has stones of low grade "a" or high grade "a + s", in proportions 1 — p and p respectively, "5" is
fixed, "p" is unknown, and "a" varies from site to site. Based on samples of scattered stones, select
the area of highest potential value for intensive search.
Note that the twopoint distribution has been used extensively by Hald and Thyregod [4] as a prior
distribution for Bayesian sampling plans. The parameters of the prior are usually estimated from
previouslyinspected lots (see pp. 36~37 of Hald and Thyregod [4]), with each such lot being assumed
to have the same probability distribution. When lots come from several sources, considerations related
to those of the present paper may be important, but have not yet been explored.
4. THE BECHHOFERSOBEL NONPARAMETRIC PROCEDURE
The nonparametric procedure ^ B s proposed by Bechhofer and Sobel (1958) takes n independent
vectors of observations. Then the largest observation in each vector is assigned a 1, and all other
observations are assigned 0. (Ties are broken at random in the assignment.) The population with the
most l's is selected as best (ties being broken at random). Let Pi be the probability that an observation
from 7Tj exceeds observations from tt\, . . ., 7r,i, 7r, +1 , . . ., 77^ (1 =£ i =S k) (assuming, without loss,
that ties have probability zero). Let P[i] =S . . . =£ pm denote the ranked Pi, . . ., p*. Then we have
essentially the problem of choosing the sample size n for the problem of selecting the cell with the
highest probability in a multinomial distribution. To attain P(CS) 5? P* whenever P[k] > 6*P\k\\ we
can use the tables of Bechhofer, Elmaghraby and Morse [1], calculated under the least favorable
configuration
(4.1) Pm= • • • =P[*i] = P[*]/0*.
For fixed A* (0 < \*< 1) and k, we can find p or p\k\ for our twopoint populations from the relation
742
(4.2)
E. J. DUDEWICZ AND C. FAN
P[k) = P+(lp) k .
If we select k and 6*, then P[k] is determined as 0*/(d* + k — 1) and, via (4.2), p is found. Note that
(4.2) will have either no, one, or two solutions (depending on 6* and k).
The following table illustrates the relative efficiency of^w. 0> B s for various P*, k, 6*,K*. The
RE is the sample size needed by 9>m divided by the sample size needed by @*bs (Note that two values
per cell can arise when (4.2) has two solutions for p. All RE' s are ratios of the smallest integer sample
sizes which guarantee the probability requirement.)
RE= (Sample Size for ^m)/ (Sample Size for ^ BS )
k
2
\ x *
e*
p* \
0.1 + 0.2 + 0.3 +
0.4 + 0.5 + 0.6 +
0.7 + 0.8 + 0.9+
0.99
3.42 1.42
0.79 0.53 0.37
0.32 0.21 0.21
3.00
0.90
11.43 2.86 1.00
0.71 0.29 0.29
0.29 0.29 0.29
0.80
10.00 1.67 1.33
1.00 0.67 0.67
0.67 0.67 0.67
0.9995
1.82
1.18 0.73 0.45
0.45 0.36 0.36
0.999
4.44 2.22
1.11 0.67 0.56
0.33 0.33 0.33
10.00
0.99
4.00 2.00
1.00 0.40 0.40
0.40 0.40 0.40
0.90
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
0.80
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
k
3
4
\ X *
e*
P* \
0.3 + 0.6 +
0.9 +
0.3 +
0.6 + 0.9 +
0.9995
1.85, 2.85 0.38, 0.69
0.31, 0.38
1.60, 3.33
0.47, 0.80 0.27, 0.33
0.999
1.82, 3.09 0.45, 0.82
0.36, 0.36
1.54, 3.38
0.38, 0.77 0.31, 0.38
10.00
0.99
1.43, 2.86 0.57, 0.71
0.43, 0.43
1.56, 3.00
0.44, 1.25 0.33, 0.44
0.90
1.33, 2.33 0.67, 0.67
0.67, 0.67
1.00, 2.50
0.50, 0.50 0.50, 0.50
0.80
1.00, 1.00 1.00, 1.00
1.00, 1.00
1.33, 1.33
0.67, 0.67 0.67, 0.67
The following table illustrates the RE oi&s vs.&Bshr various P*, k, 8*. (For 0* = 3.OO, k = 2 only
the values 0.36, 0.57, 0.67 are available, for P* =0.99, 0.90, 0.80 respectively.)
NONPARAMETRIC SELECTION
743
RE = {Sample Size for ^^/(Sample Size for £P BS )
0*= 10.00
\ k
2
3
4
p* \
0.9995
0.45,
6.55
0.38, 5.77
0.40, 5.20
0.999
0.44,
7.22
0.36, 6.18
0.38, 5.46
0.99
0.60,
8.80
0.43, 6.57
0.44, 5.33
0.90
0.67, 7.67
0.50, 6.00
0.80
0.67, 5.67
Under normal and uniform alternatives, Dudewicz (1971) found efficiencies for 3P B s vs. &m, and for
SPbs vs. procedures specially designed for uniform or normal populations. These were between 0.27
and 0.74 (i.e. in those cases the nonparametric procedure S^bs always required from 30% to 300%
large samples). We have found efficiencies for SPbs, in the case of 2point populations, which range
from 0.21 to 11.43. This shows that (as was conjectured) 2point populations furnish a more extreme
alternative, and that for some populations the nonparametric procedure 3?bs can be much more effi
cient than reasonable parametric alternatives.
5. A NEW NONPARAMETRIC PROCEDURE
The nonparametric selection procedure 8?bs of Bechhofer and Sobel [2] took n vectors of observa
tions (Xjj, . . ., Xkj) (j= 1,2, . . . , n). In each vector the largest observation was assigned a 1 (the
others 0's), and the population which amassed the most l's was selected. Tables which could be used
to implement SPbs were given by Bechhofer, Elmaghraby and Morse [1]. Dudewicz [3] showed that 8P tiS
has reasonable efficiency relative to certain alternatives, and (in Section 4 of this paper) the present
authors found results indicating that ^ B s may be surprisingly good in certain instances.
Now, replacing (Xij, . . ., X kj ) by (0, 0, . . ., 0, 1, 0, . . ., 0) (the 1 replacing the largest of
Xij, . . .,Xkj) intuitively seems a great loss of information. If instead one replaced the largest observa
tion by k, the next largest by k — 1, . . ., the smallest by'l, and selected that population achieving
the largest rank sum, how would one do? It seems intuitively clear to us that this procedure should
be better than Pas (at least under mild restrictions). It is not clear that this new procedure (^,v, say)
has an easilydetermined least favorable configuration; @bs did. (Considering the observations vector
be vector helps avoid least favorable configuration problems found by Rizvi and Woodworth [7]. These
difficulties invalidated most thenexisting nonparametric selection procedures except 3P H s and some
"subsetselection" procedures.) The use of @s for subset selection has recently been studied by
McDonald [5, 6]. The present authors and Dr. Gary C. McDonald are now studying ^v in detail in terms
of efficiency, tables needed for implementation, comparisons with other nonparametric procedures,
and modifications for other problems, and hope to present these considerations in a later paper.
REFERENCES
[1] Bechhofer, R. E., S. Elmaghraby, and N. Morse, "A singlesample multipledecision procedure
for selecting the multinomial event which has the highest probability," Annals of Mathematical
Statistics, Vol. 30, pp. 102119 (1959).
744 E J DUDEWICZ AND C. FAN
[2] Bechhofer, R. E. and M. Sobel, "Nonparametric multipledecision procedures for selecting that
one of k populations which has the highest probability of yielding the largest observation,"
Abstract, Annals of Mathematical Statistics, Vol. 29, p. 325 (1958).
[3] Dudewicz, E. J. "A nonparametric selection procedure's efficiency: largest location parameter
case," Journal of the American Statistical Association, Vol. 66, pp. 152 — 161 (1971).
[4] Hald, A. and P. Thyregod, "Bayesian single sampling plans based on linear costs and the Poisson
distribution," Matematiskfysiske Skrifter udgivet af Det Kongelige Danske Videnskabernes
Selskab, Vol. 3, No. 7, pp. 1100 (1971).
[5] McDonald, G. C. "Some multiple comparison selection procedures based on ranks," Sankhya,
Series A, Vol. 34, pp. 5364 (1972).
[6] McDonald, G. C. "The distribution of some rank statistics with applications in block design se
lection problems," Research Publication GMR1209, Research Laboratories, General Motors
Corporation, Warren, Michigan (1972). To appear, Sankhya.
[7] Rizvi, M. H. and G. G. Woodworth, "On selection procedures based on ranks: counterexamples
concerning least favorable configurations," Annals of Mathematical Statistics, Vol. 41, pp.
19421951 (1970).
SIMPLIFIED ESTIMATES OF THE PARAMETERS OF THE DOUBLE
EXPONENTIAL DISTRIBUTION BASED ON OPTIMUM ORDER
STATISTICS FROM A MIDDLECENSORED SAMPLE
M. Ahsanullah
Food and Drug Directorate, Ottawa
and
M. A. Rahim*
Statistics Division
Indian and Northern Affairs
Ottawa, Canada
ABSTRACT
In an ordered sample from a given population, a few of the consecutive observations
from somewhere in the middle may be missing. Further, we may be constrained to use a
few, and not all, of the remaining observations for purposes of estimation of population
parameters. In this paper, such a situation is considered for the double exponential distribu
tion and best linear unbiased estimates are obtained for its parameters, based on a choice of
an optimum set of order statistics when the number of observations in the set are prefixed.
1. INTRODUCTION
The density function of the double exponential distribution is given by
(l.i) fx(x,e 1 ,e t ,) = — e e * , oo<^<oo
O<0 2 .
The problem of estimating the parameters 0\ and #2, based on order statistics, have been studied in
the past under various situations.
Sarhan [4] gave the best linear unbiased estimate (BLUE) of Qi and 02 based on order statistics
when the original sample was of size n = 5 and the available sample, after censoring, was n' = 2(1)5.
He considered left or right censored samples and used all the n' observations. Govindarajulu [2]
extended these results for n = 20 and n' = 2(1)20 when censoring was done symmetrically from left
and right. He also used all the n' observations. Chan and Chan [1] obtained BLUE of 6\ and 2 based
on a choice of k(^ n) optimum order statistics for n= 1(1)20 and £(=£ n) = 1 (1)4. He considered the
case when full sample n was available.
There are, however, practical situations, when the censoring occurs from somewhere in the
middle of an ordered sample. Sarhan and Greenberg [5] mentioned the case of telemetry where signals
are sent at regular intervals and a few may be missing during the journey. Middlecensored samples
may also occur due to failure of the measuring instrument to record observations or due to offshifts
This research was performed when the author was at Carleton University, Ottawa, Canada.
745
746 M AHSANULLAH AND M. A. RAHIM
or weekend interruptions during the course of an experiment — particularly when the variate under
observation is a time period (i.e., the period to failure of a piece of equipment undergoing testing;
survival period of bacteria, etc.).
From such a middle censored sample, one may further decide — for reasons of economy or practical
convenience — to use a few, and not all of the available observations. The question then arises, which
of the available observations to use?
In this paper we consider such a situation when we have an ordered sample of size n,
X{\) < X(2) < . . ■ < X(H,) < X(R 2 ) < Jt(« 2+ i) < . . . <*(„),
where Ri and R 2 are some integers such that 1 «£ /?i < R 2 =£ n and the middle observations *<«, + />,
Jf(«,+2), . . . X(h 2 d, are missing. For a given k = ki + k 2 , such that k\ < R\ and k 2 < n —  (R 2 — 1), our
objective is to determine the optimum ranks «", n", . . . n% , njj , . . . nJJ i+A . 2 , such that
1 *£ n? < ra» < < n° ki ^ R t
^2^< fcl+1) << 1+2) < .... <n° {ki+k2) *£n
and obtain BLUE of 8\ and 6 2 based on the observations
*(„o), *(„;), . . . X(n»J, X(n° k ^) . . . *( n J^).
2. OPTIMUM RANKS AND "BLUE" OF 0, AND B 2
Consider any set of ranks «i, n 2 , . . . . rat,, n* 1+ i . . n ki + k2 such that
1 =£ Hi < n 2 < . . . . < n*,*£ R U R 2 ^ n k>+1 < n ki+2 < . . . <n kl+k2 ^n.
Then, for the double exponential distribution (11), transforming to standardized variable z, so that
x = 6i + 6 2 Z, we can write
(2.1) E(x (ni )) = e i + e 2 E(z u , i) ), £=1,2 .... kukt + 1, . . . k t + k 2
V(x (ni) )=dlV(z {n0 )
Cov (*(„,), X(nj)) = 6\ Cov (Z(„j), Z(n } )).
Hence, by applying the generalized least square theorem and following Lloyd [3], the BLUE's of d x
and 2 can be obtained in three different situations, namely, when 9 X = 0J> is known, or 6 2 = 6% is known,
or 6\ and 6 2 are both unknown. These best linear unbiased estimates and their variances and gen
eralized variance are shown below, which we have expressed in terms of following matrices
y =[*(»!,), *(»,), .... X(n kl + n k2 )], a'=[E(z(» l )),E(z l „ 2 )), . . . E(z(„ ki+k2 ))
DOUBLE EXPONENTIAL DISTRIBUTION
1'= [1,1, ... . 1], V=[Co\ (zinihZ(„j))].
CASE 1: 0i = 0? is known:
The BLUE of 02 denoted by 02 and its variance V(6 2 ) are given by
« _ a'V 1 Y 0?(aT'l)
(22) d2 ~a'V 1 a a'V^a '
= b( ni )X(n,)+ b(n 2 )X(n 2 )+ . . . + b(n kl+k JHn kl+k J — 60?;
02
(23) ^) = (?Fm
747
CASE 2: 02=02 is known:
The BLUE of 0i denoted by 0i and its variance V(di) are given by
h (lT'K) 2 °(lT'«)
(2.4) »i (rr ,D (i'Kil)
= a(n,)X(„ 1 )+a(n 1! )a;( n2 )+ .... + a (n fei+fc2 )*(n fci+fcj ) — a0§;
(2.5) r(0,) = 0i/(lT 1 l).
CASE 3: 0i, 02 both unknown:
The BLUE's of 0i, 02 denoted by 0i, 02 and their generalized variance GV(d\ 2 ) are given by
(2.6) 6i =  aTYJ=V'{\a'  a\')^, ±={VVn)(ot'V l a) (VV*a)*
2 = ITT
or
01 = a ( „ l) aC < „ 1 ) + a (n^(n,)+ .... +a(n kl + k2 )X(n kl + kt )
02 = 6(n 1 )*(n 1 )+6(„,)X(„ ]t )+ .... +b(n kl + k2 )X(n kl + kj ).
(2.7) GF(0,02) = r(0 1 )r(0 2 )(Cov (0., 2 )) 2
= 0^/[(a'r 1 a)(lT 1 l)(lT 1 a) 2 ].
748 M AHSANULLAH AND M. A. RAHIM
Now, for a given Y the 'a' and '6' coefficients in the above expressions for BLUE's are determinable
by direct matrix multiplication, when a and V are known, i.e., the quantities E{z(n t )), ^(z(n 1 )),Cov(z(u i .)),
Z(tij)) are known. In fact, these quantities have been tabulated by Govindarajulu [2]. Hence, using
those values; for any fixed n, truncation points R\, R 2 , and A: ; it is possible to take all combination of
ranks and compute the values of ^(^1), ^(^2), or GV(0i 2 ), as the case may be. We then take that set
of ranks as optimum ranks for which the variance/generalized variance is minimum and denote them as
«j, n°, n", .... « ( ° fcl) , "(°fr 1+ i)i .... "(°fc 1+ fc 2 ) Corresponding to these optimum ranks the BLUE's
are obtained by computing the 'a' and 'ft' coefficients by direct matrix multiplication from the results in
[2.2], [2.4], and [2.6]. These computations have been carried out and the optimum ranks and corre
sponding BLUE's have been obtained for n = 3(l)10, k = 2(1)5, and for all possible truncation points
Ri and #2.
The results have been tabulated for the aforesaid three cases separately, showing the optimum
ranks as well as the corresponding coefficients of the BLUE, in a readily usable form. A part of this
tabulation is presented in Table 1 and an example follows showing how it can be used for obtaining
the estimates of 0i and 02 as well as the efficiency of such estimates (full tables for n — 3(1)10, and
k = 2(1)5, are available from the authors).
3. AN EXAMPLE
Consider a censored sample *(u, X(2), x<3), *(6), where X( 4 ) and £(5) are missing. Further, suppose
we want to use three of the available observations for estimating unknown 0i and 02. From Table 1,
for 71 = 6, /?i = 3, R 2 — 6, k — 3, we immediately read out the optimum ranks as n°=l, n\ — ^, ^3 = 6.
The corresponding BLUE's are obtained from the table as —
0i = a { i)X { i) + a (3 )JC( 3 ) + a(6)At(6) = 0.0021 %(\) + 0.8856 X( 3 ) + 0.1124 * (6 )
02 = b ( i)X( i) + 6(3)*( 3 ) + 6(6)^(6) = — 0.2644*0) — 0.0362x (3) + 0.3006 Jt (6 ).
Also— GF(0, 2 ) = (0.0633). If we used all four observations we would have (from Table 1, fc=4»),
2
1 „ .
— GV(did 2 ) = (0.0599). Hence, efficiency of our estimates compared to full available sample is 0.94,
"2
which is quite high.
ACKNOWLEDGEMENT
The authors record their deep gratitude to Prof. A. K. Md. E. Saleh Dept. of Mathematics, Carleton
University, Ottawa, who suggested the problem and rendered assistance during the work.
The authors are also thankful to the referee for helpful comments.
DOUBLE EXPONENTIAL DISTRIBUTION
749
TABLE 1. Showing Coefficients of the BLUE's of the Parameters of Double Exponential Distribution
with kOptimum Order Statistics from Samples Censored in the Middle (shown here for n = 4,5,6,7,
and k = 3,4 only)
For k = 3
n
i
*2
ii
Hi
n 3
1
3
3
\rtto
igrft)
1 . .
Cov(0„ t )
"2
1 . .
GV(e u e 2 )
2
4
1
3
1
3
4
11 
0.2103
0.3849
0.7707
0.0639
0.0191
0.3210
0.3127
0.4782
0.0297
0.1487
4
2
4
1
2
4
0.0191
0.3210
0.7707
0.0639
0.2103
0.3849
0.3127
0.4782
0.0297
0.1487
5
1
3
1
3
5
0.0670
0.3148
0.8660
0.0000
0.0670
0.3148
0.2547
0.3395
0.0000
0.0865
5
1
4
1
4
5
0.2529
0.3436
0.7729
0.0903
0.0258
0.2533
0.2496
0.4515
0.0439
0.1108
5
2
4
1
2
4
0.0119
0.3063
0.4776
0.2950
0.5105
0.6013
0.3009
0.3583
0.0053
0.1078
5
2
5
1
2
5
0.0258
0.2533
0.7729
0.0903
0.2529
0.3436
0.2496
0.4515
0.0439
0.1108
5
3
5
1
3
5
0.0670
0.3148
0.8660
0.0000
0.0670
0.3148
0.2547
0.3395
0.0000
0.0865
6
1
3
1
3
6
0.0021
0.2644
0.8856
0.0362
0.1124
0.3006
0.2162
0.2948
0.0217
0.0633
6
1
4
1
4
6
0.1124
0.3006
0.8856
0.0362
0.0021
0.2644
0.2162
0.2948
0.0217
0.0633
6
1
5
1
5
6
0.2820
0.3139
0.7612
0.1019
0.0431
0.2120
0.2104
0.4481
0.0504
0.0917
6
2
4
1
4
6
0.1124
0.3006
0.8856
0.0362
0.0021
0.2644
0.2162
0.2948
0.0217
0.0633
6
2
5
2
5
6
0.5035
0.5096
0.4912
0.2754
0.0053
0.2342
0.2320
0.3369
0.0016
0.0782
6
2
6
1
2
6
0.0431
0.2120
0.7612
0.1019
0.2820
0.3139
0.2104
0.4481
0.0504
0.0917
6
3
5
1
3
6
0.0021
0.8856
0.1124
0.2162
0.2948
0.0217
0.0633
6
3
6
1
3
6
0.2644
0.0021
0.0362
0.8856
0.3006
0.1124
0.2162
0.2948
0.0217
0.0633
6
4
6
1
3
6
0.2644
0.0021
0.2644
0.0362
0.8856
0.0362
0.3006
0.1124
0.3006
0.2162
0.2948
0.0217
0.0633
7
1
3
1
4
7
0.0383
0.9233
0.0383
0.1911
0.2316
0.0000
0.0443
7
7
1
1
4
5
1
1
4
5
7
7
0.2632
0.0383
0.2632
0.1483
0.0000
0.9233
0.0000
0.8810
0.2632
0.0383
0.2632
0.0293
0.1911
0.1889
0.2316
0.2837
0.0000
0.0348
0.0443
0.0524
7
1
6
1
6
7
0.2859
0.3027
0.0568
0.7477
0.2291
0.0504
0.1835
0.4512
0.0534
0.0799
7
2
4
1
4
7
0.2916
0.0383
0.1074
0.9233
0.1842
0.0383
0.1911
0.2316
0.0000
0.0443
7
2
5
2
5
7
0.2632
0.3006
0.0000
0.6981
0.2632
0.0013
0.1980
0.2520
0.0092
0.0498
7
2
6
1
2
6
0.4962
0.0023
0.1920
0.2556
0.4963
0.2581
0.2406
0.5013
0.4501
0.1911
0.3286
0.0006
0.0628
750
M. AHSANULLAH AND M. A. RAHIM
Table 1. Showing Coefficients of the BLUE's of the Parameters of Double Exponential Distribution
with kOptimum Order Statistics from Samples Censored in the Middle {shown here for n = 4,5,6,7,
and k = 3,4 only) —Continued
For A = 3
n
«,
«2
«i
n 2
«3
a <"°>
fl(n°)
3
^(5.)
k™
1 . .
Cov(0„ 2 )
"2
GVCduh)
e*
7
2
7
1
2
7
0.0504
0.1842
0.7477
0.1074
0.3027
0.2916
0.1835
0.4512
0.0534
0.0799
7
3
5
2
5
7
0.3006
0.4962
0.6981
0.2556
0.0013
0.2406
0.1980
0.2520
0.0092
0.0498
7
3
6
3
6
0.0013
0.2406
0.6981
0.2556
0.3006
0.4962
0.1980
0.2520
0.0092
0.0498
7
3
7
3
7
0.0293
0.2291
0.8810
0.0568
0.1483
0.2859
0.1889
0.2837
0.0348
0.0524
7
4
6
4
7
0.0383
 0.2632
0.9233
0.0000
0.0383
0.2632
0.1911
0.2316
0.0000
0.0443
7
4
7
4
7
0.0383
0.2632
0.9233
0.0000
0.0383
0.2632
0.1911
0.2316
0.0000
0.0443
7
5
7
4
7
0.0383
0.2632
0.9233
0.0000
0.0383
0.2632
0.1911
0.2316
0.0000
0.0443
For k —
4
n
«,
Rt
n t
"2
«3
n 4
"(no)
6(no)
2
«(n«)
6(no)
3
a(„o)
irA)
1
1 . 
Cov(0,,0 2 )
2
1 , 
GF(«,,«0
2
5
1
3
3
4
5
0.0811
 0.2991
0.7021
0.1821
0.2123
0.2357
0.0046
0.2455
0.2413
0.3286
0.0121
0.0792
5
2
4
2
4
5
0.0173
 0.2331
0.4827
 0.2264
0.4827
0.2264
0.0173
0.2331
0.2290
0.3579
0.0000
0.0819
5
3
5
2
3
5
0.0046
 0.2455
0.2123
 0.2357
0.7021
0.1821
0.0811
0.2991
0.2413
0.3286
0.0121
0.0792
6
1
3
1
3
4
6
0.0273
 0.2491
0.4727
 0.2861
0.4727
0.2861
0.0273
0.2491
0.2031
0.2590
0.0000
0.0526
6
1
4
4
5
6
0.1173
 0.2873
0.8223
0.1331
0.0794
0.2123
 0.0190
0.2081
0.2063
0.2934
 0.0254
0.0599
6
2
4
2
4
6
0.0069
0.1974
0.2548
 0.2493
0.7111
0.2070
0.0272
0.2398
0.1956
0.2732
 0.0005
0.0534
6
2
5
2
5
6
0.0069
0.1893
0.4931
 0.2234
0.4931
0.2234
0.0069
0.1893
0.1875
0.3368
0.0000
0.0632
6
3
5
3
5
6
0.0272
 0.2398
0.7111
 0.2090
0.2548
0.2493
0.0069
0.1974
0.1956
0.2732
0.0005
0.0534
6
3
6
2
3
6
 0.0190
 0.2081
0.0794
 0.2123
0.8223
0.1331
0.1173
0.2873
0.2063
0.2934
0.0254
0.0599
6
4
6
3
4
6
0.0273
 0.2491
0.4727
 0.2861
0.4727
0.2861
0.0273
0.2491
0.2031
0.2590
0.0000
0.0526
7
1
3
3
5
7
0.0120
0.2044
0.4880
 0.2917
0.4880
0.2917
0.0120
0.2044
0.1681
0.2254
0.0000
0.0379
7
1
4
4
6
7
0.0497
 0.2407
0.8385
0.1686
0.1177
0.2341
 0.0059
0.1752
0.1744
0.2273
 0.0084
0.0396
7
1
5
5
6
7
0.1491
 0.2750
0.8698
 0.0939
0.0139
0.1866
 0.0327
0.1823
0.1815
0.2836
 0.0353
0.0502
7
2
4
2
4
7
 0.0059
0.1752
0.1177
 0.2341
0.8385
0.1686
0.0497
0.2407
0.1744
0.2273
0.0084
0.0396
NONPARAMETRIC SELECTION
751
TABLE 1. Showing Coefficients of the BLUE's of the Parameters of Double Exponential Distribution
with kOptimum Order Statistics from Samples Censored in the Middle (shown here for n = 4,5,6,7,
and k= 3,4 only) —Continued
For k = 4
R
R,
«2
n,
"2
«3
n 4
6 (no)
a (n o)
O(no)
6(n0)
a<„o)
b{ n o)
1
— V(0 2 )
9 2
k rA >
1 . .
~Cov(0,,0 2 )
l
1 . .
7
2
5
2
5
7
0.0099
0.1658
0.2858
 0.2481
0.7006
0.2134
0.0037
0.2005
0.1650
0.2519
 0.0072
0.0415
7
2
6
2
6
,7
0.0029
0.1610
0.4971
 0.2166
0.4971
0.2166
0.0029
0.1610
0.1602
0.3286
0.0000
0.0526
7
3
5
3
5
7
0.0120
 0.2044
0.4880
 0.2917
0.4880
0.2917
0.0120
0.2044
0.1681
0.2254
0.0000
0.0379
7
3
6
3
6
7
0.0037
 0.2005
0.7006
 0.2134
0.2858
0.2481
0.0099
0.1658
0.1650
0.2519
0.0072
0.0415
7
3
7
2
3
7
 0.0327
0.1823
0.0139
0.1866
0.8698
0.0939
0.1491
0.2750
0.1815
0.2836
0.0353
0.0502
7
4
6
2
4
7
 0.0059
0.1752
0.1177
 0.2341
0.8385
0.1686
0.0497
0.2407
0.1744
0.2273
0.0084
0.0396
7
4
7
2
4
7
 0.0059
0.1752
0.1177
 0.2341
0.8385
0.1686
0.0497
0.2407
0.1744
0.2273
0.0084
0.03%
7
5
7
3
5
7
0.0120
0.2044
0.4880
 0.2917
0.4880
0.2917
0.0120
0.2044
0.1681
0.2254
0.0000
0.0379
REFERENCES
[1] Chan, L. K. and N. N. Chan, "Estimation of the Parameters of the Double Exponential Distribution
Based on Selected Order Statistics," Bulletin of the Institute of Statistical Research and Training,
Dacca University, Vol. 3, No. 2, pp. 2140 (1969).
[2] Govindarajulu, Z., "Best Linear Estimates Under Symmetric Censoring of the Parameters of a
Double Exponential Population," J. Am. Stat. Assoc, Vol. 61, pp. 248258 (1966).
[3] Lloyd, E. H., "Least Squares Estimation of Location and Scale Parameters Using Order Statistics,"
Biometrika, Vol. 39, pp. 8895 (1952).
[4] Sarhan, A. E., "Estimation of the Mean and Standard Deviation by Order Statistics," Annals of
Mathematical Statistics, Vol. 25, pp. 317 (1954).
[5] Sarhan, A. E. and B. Greenburg, "Linear Estimates for Doubly Censored Samples from the Ex
ponential Distribution with Observations also Missing from the Middle," Bulletin of the Inter
national Statistical Institute, 36th session, Vol. 42, Book 2, pp. 1195 1204 (1967).
THE SINGLE SERVER QUEUE IN DISCRETE TIMENUMERICAL
ANALYSIS IV
David Heimann* and Marcel F. Neutst
Purdue University
ABSTRACT
The nonlinear difference equation for the distribution of the busy period for an un
bounded discrete time queue of M  G  1 type is solved numerically by a monotone iterative
procedure. A starting solution is found by computing a first passage time distribution in a
truncated version of the queue.
1. INTRODUCTION
We consider the single server queue in discrete time, studied by Dafermos and Neuts [1]. The
numbers of customers forming the queue during successive units of time form a sequence of independ
ent, identically distributed random variables with common probability density {p v }, where *S v < K.
In numerical investigations, we assume that K is a finite positive integer. The service times of the
successive customers are independent, identically distributed random variables with common density
{r„}, where 1 ** v ss L 2 . In numerical investigations, we assume thatZ/2 is finite. The numbers of arrivals
during successive units of time and the successive service times are assumed to be independent se
quences of random variables.
In this paper we shall consider the case of an unbounded queue, in which there is no upper bound to
the number of customers which may be present at any time, and also the case of a bounded queue. In
the latter case, there is an upper bound Li to the number of customers present at any one time. All
customers arriving when the queue length is L x are lost. For notational simplicity, we write L\ = + °°,
for the unbounded queue.
It is known [1, 3] that both cases may be studied in terms of a bivariate Markov chain {(X n , Y„),
n?0}, where X„ denotes the number of customers present at time n+ and Y„ denotes the residual
service time of the customer in service at time n +. We note that X n = 0, if and only if Y„ = 0. The
Markov chain {(X n , Y„), n ^ 0} has as its state space, the point (0, 0) and all pairs (i,j), where
l«i« L,,l^;'s£L 2 .
For purposes of easy reference, we recall that there are four possible types of transitions during
a unit of time in the Markov chain { (X„, Y n ) , n 2* 0} :
(1) When X n «* 1 , Y„ > 1 the residual service time Y n +\ = Y„ — l, and only arrivals to the queue can
occur. Z„+i = min (L x , X„+ v), where v denotes the number of arrivals in the time interval (n, n + 1).
* The research of this author was sponsored by the Purdue Research Foundation through an XR Grant.
t Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No.
AFOSR 722331. The United States Government is authorized to reproduce and distribute reprints for governmental purposes
notwithstanding any copyright notation hereon.
753
754 D. HEIMANN AND M. F. NEUTS
(2) When X n > 1, Y„ = 1, a departure occurs at time re + 1 and the next customer in line begins
service. The residual service time Y„ + i is equal to the service time of the customer whose service begins
at time n + 1.
(3) When X n = Y n = 1, a departure occurs at time re + 1. At time n + 1, the queue may become
empty if there are no arrivals during the time interval (re, n + 1) . If there are arrivals during (re, re + 1 ) ,
a new service is initiated at time re + 1.
(4) When X n =Y n = 0, the queue is empty at time re. A new service and also a new busy period
is started at time /i + 1, if and only if there are arrivals during (re, re + 1).
2. THE BUSY PERIOD FOR THE UNBOUNDED QUEUE
We consider a queue with one customer at time re — 0, whose service is just beginning. We denote
by /3 n the probability that the queue becomes empty for the first time at time re. The sequence {/3 n }
is the probability density of the duration of the busy period. Since by assumption each customer
requires at least one unit of time for service, j3<> = 0.
Denoting by {p^ 1 } and {^, fc) }, respectively, the Afold convolutions of the probability densities
{p v } and {/"„} and setting p ( „ 0) = r < 1/ 0) = 8 „, the sequence {/3«} satisfies the nonlinear difference equation
j=l r=0
for re > 1. Equation (1) is obtained by application of the law of total probability. The probability that the
first service lasts iorj units of time and that v customers arrive during it is ryp^. If v ¥" 0, these v customers
may be considered as the initial customers of v independent busy periods. The probability that these
v busy periods together take up exactly the re — j remaining units of time is given by j3 ( „"Jj. Equation
(1) is recursive, since for each re, the expression on the right involves only j8i, . . ., /3«i and the
terms of the convolutions of the sequence {/S^} with an index not exceeding re — 1.
Nevertheless, direct numerical computation of the recurrence relation (1) requires a prohibitive
amount of computer memory which seriously limits its practical usefulness.
We note that the second summation in (1) involves only a finite number of terms, since p ( ji is
zero for v > Kj. Even for j= L 2 , we need to consider only v such that ^ v ^ L 2 K.
Defining
(2) a (j) = r jP l, {oil^j^L 2 ,
= • elsewhere
and
oAj) = r jP [J\ for 1 < ; < Li, 1 < v < Kj,
= 0, elsewhere,
we may write Equation (1) in the general form
SINGLE SERVER QUEUE IV 755
(3) {/34 = 2{a,} *{#>}.
The nonlinear difference Equation (1) is therefore of the form
(4) B = P(B),
where P(B) is a polynomial a + ai * /3 + a 2 * /3 (2) + . . . + a K L 2 * B (KU) in the convolution algebra
over the sequences with nonnegative indices. We shall show below that the Equation (4) may be
conveniently solved by successive substitutions. We shall then concentrate on the problem of selecting
a good starting sequence for this iterative solution.
Before doing so, we shall derive an explicit expression for the probabilities B n , which is analogous
to the explicit formula for the distribution of the busy period in the M \ G  1 queue, due to L. Takacs
[4].
Denoting the probability generating functions of {Bn}, {pn}, and {r n } by B{z), P(z), and R(z),
respectively, Equation (1) may be written as:
(5) B{z)=R[zP[B(z)]], z<l.
The following theorem is wellknown and may be proved by a standard method involving Rouche's
theorem.
THEOREM 1: For every \ z\ < 1, the functional equation
(6) t = R[zP{0],
has a unique root in the unit disk  £  < 1. This root C = B(z) is an analytic function of z, which is
continuous on the boundary  z  = 1. The quantity = 5(1—) is the smallest positive root of the
equation
(7) i = R[P(C)]
Moreover = 1, if and only if P'(l )«'( 1 ) ^ 1, and < d < 1, if and only if P' (1 )«'(1 ) > 1.
If £ = P'(l _ )/?'(!—) < \ t we sa y that the queue is stable. The irreducible, aperiodic Markov
chain {(X„, Y„), n*z 0} is then positive recurrent. If £= 1, we say that the queue is critical; the
Markov chain is then nullrecurrent. If £ > 1, the queue is unstable. The probability that the busy
period does not end is 1 — > 0.
For a stable queue, the busy period has moments of all orders. Klimko and Neuts [2] have shown
that moments up to order 50 may be computed numerically by repeated applications of Faa di Bruno's
formula.
From Equation (6), an explicit formula for B{z) may be derived by use of Lagrange's Inversion
Formula [6]. We first modify the functional Equation (5) by means of the inverse function R x (•). For
every £,  £  < 1, R(0 is defined as that branch of the solution of R[R l (0]= £* which is real for
756 D. HEIMANN AND M. F. NEUTS
*£ £ < 1. /? _1 (  ) is then analytic in  (  < 1. Setting R~ l [B(z)]= w(z), Equation (5) may be rewritten
as:
(8) w(z)=zP[R(w(z))],
for  z  *£ 8 sS 1. Again by Rouche's theorem, we obtain that the functional equation
(9) £ = z/>[fl(£)], *N8<1,
has a unique solution inside  £  ^ 1, for every z, with \z\ ^8. The function B{z) is then given by
(10) B(z)=R(0=R[w(z)].
Applying Lagrange's Inversion Formula to (9), it follows that for all z with  z  *£ 8, we have
(ID flU)=«(0) + ^[^{«'(5)[P[/?( 5 )]]"}} so .
We recall that R(0) = 0, so that /3o = 0. For n 3= 1, we have
by Leibniz's Rule. We note that
(13) [ dF* l =(l/+1)!r " +1 ' forO«y<I,l,
and zero for t> > L 2 , so that Formula (12) may be rewritten as:
(14) *I;g (n,)r B .„ I[>[*(,)]] (
l«=o
The derivatives of P"[/?(s)] are given by Faa di Bruno's Formula as follows:
(15) \£ P n [R(s)]] = ± T^p 2 TX^S"^ 1 • • • *•
Ji+... + vjv=v
SINGLE SERVER QUEUE IV 757
for 1 s£ v *£ n, where,
(161 «  1 \ d'P[K(sm *
for i 3= 1, since 0, is the coefficient of s ! in the power series expansion of P [R(s) ]. Combining Formulas
(14), (15), and (16) we obtain the following explicit expression for B n , n 2= 1:
a?) 0. = r„p o + (»i)!*£ (»^. 2 7^7)7 2 0'f['
" = 1 '=1 j, + ••+/'„=<
jj + 2j 2 + ... + !>,;„ =v,
where 0; is given by (16).
Although for every n, the right hand side in Formula (17) is a finite sum, it is not a suitable expres
sion for numerical computation. The number of terms in it grows very rapidly for higher values of n.
Using the efficient algorithm developed by E. Klimko [2] for generating the indices ju i = 1, . . .,vin
the last sum, one can compute at most the first 50 terms of the sequence {/3„} without running into
prohibitive computation times and memory storage requirements.
3. THE ITERATIVE PROCEDURE
The fixed point Equation (4) suggests that the nonlinear difference equation for the sequence
{fin} may lend itself well to a solution by successive substitutions. We shall prove that this is the
case and furthermore that it is possible to find a sequence of solutions which converges monotonically
to the density {B„}, in a sense to be defined below.
We shall consider the set & of all sequences b.= {b„, n 3= 0} for which b„ 2=0 for all n 2= 0, and
2 b n < 1. The convolution product b * c of any two sequences in 88 is itself in 38. We denote by b (k) the
n =
Mold convolution of 6_ with itself and we define b { 0) to be the sequence (1 , 0, 0, . . . ). n
To every sequence in S8 we associate the sequence B = {B„, n 2* 0}, where B„ = 2 b v , for n 2* 0.
£
We further define the partial ordering •< on the set ^, as follows:
£
(18) b_ < _c if and only if B„ *£ C„, for all n 2= 0.
It is straightforward to verify that if a, b, ce&, then
£ £
(19) a L < b_ implies a_* c_< b_* £■
£ £
LEMMA 1: If a,b_€@, then a. < b implies a< fc > < 6<*>, for k 2= 0.
PROOF: The statement is clearly true for k =S 1. Assuming it to hold for any k, we have
a (fc+D = a (*) * a < &.<*> *a=a * b (k) < b_*A (k) = h ik+1) 
758 D. HEIMANN AND M. F. NEUTS
The lemma holds therefore by induction on k.
We now consider a sequence of densities v a, v 5* 0, belonging to ^ such that
t>=0
and we form the convolution series
(20) A(b) = J ,o*6f").
THEOREM 2: For 6c^, „ae^, for v ^ and
x
p =
we have that A{b)e88. Furthermore b «< c imphes that A(b) < A(c).
PROOF: The terms of the sequence A(b) are clearly nonnegative. Moreover, denoting
T{b)=fb n ,
n=0
we have that T{b * c) = T(b_)T(c).
It follows that
T[A(b) ] = f T ( v a)T"(b) =£ f T(,a) *S 1.
Furthermore 6 < c_ implies that v a *i> (,,) "> >>£ * c ( "\ for all v ^ and therefore by summation on
We define the iterative process by the following relation, derived from Equation (3):
(21) ^ = 2»" **i^ < " ) ' for A 3*1,
«=o
where the sequences a* are defined in Equation (2). The sequences fc/J are the successive iterates
and the sequence q^ is the starting solution.
Since the right hand side of Equation (21) is of the form A(ki§), we use Theorem 2 to prove the
convergence of the iterative process.
£
THEOREM 3: Let {kb}, k 5= 0, be a sequence of densities such that obe&, \b_ < ob, and k+ib_ =
A(kb) for k 3* 0. Then {kb} converges monotonically to a unique limit ^e^?.
PROOF: Clearly k b€@ for £ 2* 0, by Theorem 2. Let k b' = *+,£. Then o&' < „£, and if moreover
fc6 < ^6, then by Theorem 2 A{ k V) .< /4( fc 6), or fc+16' ^ fc+i6. Therefore, by induction on k, we
SINGLE SERVER QUEUE IV 759
o
have k+ib < kb, for all k 5* 0. Convergence follows from the fact that the sequence (0, 0, . . .) is a
uniform lower bound for all of the tfe's. Because of the monotone convergence, the limit ^ is also in
38. This limit jS is unique, since clearly j8 = A({3) by passage to the limit in (21) and since Equation
(17) admits of a unique solution.
A similar monotonicity result can be obtained when ob < t b. Even without the restriction
\b < ob, we have:
THEOREM 4: If { k §}, k ^ 0, is a sequence of densities, such that k+ i@_ = A( k p) and ofSe®,
then {kfi} converges term wise to ^3.
PROOF: Consider the sequences of iterates {kb} and {kb_*} , where * + 16 = A ( k b),k + ib* = A (kb*)
for &> 0, and o£=( 1,0,0, . . .), b*= (0, 0, 0, . . .). Since ,6 < ob ando&*< xb*, Theorem 3 implies
that both sequences converge monotonically to §_. Also, since ob* 5 o§_ _• 06, we have jt6* < kfi •< kb for
all A: ^ 0. Therefore {*£} converges to^J.
REMARK: The starting solution 06 and oh* are readily available and yield monotone sequences of
approximants. However, since each iteration involves a large number of elementary operations, it is
clearly to our advantage to obtain first a starting solution closer to the sequence /J satisfying (17). Such
starting solutions may be obtained by calculating the distribution of the busy period for a bounded
queue, having the same arrival and service distributions as the unbounded queue under consideration.
4. THE BUSY PERIOD FOR THE BOUNDED QUEUE
The probability density of the busy period for the bounded queue is found by a standard investiga
tion of the absorption time distribution in a finite Markov chain. We consider a queue with upper
bound L\ starting at time n = 0, with i customers and a residual service time j, where 1 < i < L\,
l^j^ L 2 .
G n (i,j) denotes the probability that the queue with upper bound L t becomes empty no later than
time n, given the initial state (i,j) The probability gn(i, j) that the queue ends exactly at time n is
then given by the difference G„(i, j) — G„i (i,j)
If /3„(Li) denotes the probability that a queue with upper bound L x and one customer initially,
becomes empty for the first time at time n, and B n (Li) denotes the probability that such a queue
becomes empty for the first time no later than time n, then
u
(22) /8»(Li) = X 0*»0. 7'), for n?l,
and
(23) B n (L l )=2 3m(Li) = y rfinilj).
m=l y=\
In order to evaluate the probability density {(3„(Li)}, it suffices therefore to compute the conditional
probabilities G„(i,j).
The probabilities G„(i, j) satisfy the following recurrence relations:
(24) G n (iJ) = ^p v Gni[min (i + v, Li), j1],
760 D. HEIMANN AND M. F. NEUTS
fori 2= l,j> 1.
for »» 2, ./=1.
G„(i, D = 2 r>JftC»i[min(» + »Ui).i].
fc=l v =
G„(l, l)=p +2 r k ^ p„G„_i[min (i>, L,), A:]
with the initial conditions
(25) G {iJ) = 8 i0 8j .
The recurrence relations (24) are immediate by consideration of the possible states of the Markov
chain after one unit of time.
In the event relating to the probability G n (i,j), it is possible that the queue length attains the
upper bound L x before the queue becomes empty for the first time. In order to study the effect of
imposing an upper bound to the length of the queue, we shall also consider the taboo probabilities
&n(i,j), defined as follows: G n {i,j) is the conditional probability that a queue with initial state (*,/)
becomes empty no later than time n, without the queue length exceeding L\ — 1 in between.
The probabilities G„(i,j) satisfy the recurrence relations
min (K, Zjli)
(26) G„(i,j)= £ PrGniii+vJl),
for i 3 s 1 , j > 1 and
L 2 min (K, L t i)
C»(i,l)=Jr t £ p v Gni(i+vl,k),
k=l v=0
for i 3 s 2, j = 1, and
L 2 min (K, f.,1)
G„(l, 1) =Po + ^ r k 2 PvGnl(v,k),
for n 5= 1, with the initial conditions
(27) Go(i,j) = 8 io 8 j o.
For a given Li, the quantity
(28) p„(L 1 ) = 2r j [G n (lJ)G n  i (lJ)],
is the probability that a busy period with one customer initially ends no later than time n and that the
maximum queue length during that busy period is less than L u The probability (3„ (Li) is of course the
SINGLE SERVER QUEUE IV 761
same for the unbounded queue as for a bounded queue whose bound is at least L x . Similarly to Equa
tion (23), we also define
(29) £„(£,)=;£ p m (L l ) = 2r j C n (l,j).
m=l j=i
THEOREM 5: For all i u i 2 J satisfying «£ i, *£ i 2 =£ L u «j =s L 2 , we have
(30) G n (i u j)^G n (i 2 ,j),
for all n > 0.
PROOF: The inequality clearly holds for n — 0. Using formula (24) we obtain
(3D
K K
G„(i u j)= ^ p v G„i [min (ii + v, Li),j— 1] 2= ^ P* Gni [nun (i 2 + v, L 1 )J—l] = G n (i2,j),
l> = l/ =
for ii 3s l,j > 1. A similar argument holds in the cases ii > 1, i 2 > 1,7=1, and ii = l, i 2 ** 1, 7= 1.
The arguments for the cases ii=j—0, i 2 >0 and ii = i 2 =j=0 are obvious. The result of Theorem 5
also holds for the probabilities G n (i,j).
In Theorems 6, 9, and 10 below, we consider two different upper bounds L\ and L'\. In order to
indicate which upper bound is used, we write G n ,L\ (i,j) and G n ,L\ (i,j) for G„ (i,j) and G„ (i,j),
respectively, and correspondingly for L\ .
THEOREM 6: L\ *£ L'[ implies that
(32a) Gnj'SU) ^G nJA {i,j),
and
(32b) G„,, A (i,j) ^G n ,,*(i,j),
for *£ i =s L\ , s= j === L 2 , and n 3* 0.
PROOF: The proof is by induction on n. The statements are obvious when n — 0. Assume them to
hold for n — 1. We shall prove only the case where i 2* 1, j ; > 1 in detail. The other cases are similar.
The following inequalities prove (32a):
min{K,L[il) min(A,/.;il)
G«,z.;(i,j)= £ PpG u i,v t {i+v,jl)* £ P* G„_i > z, ; (i + ?,y 1)
„= «>=o
min( K,/,yi 1)
s= ^ p„ C„_i,/.j(j i + V,j 1) = G n ,i.>(i,j).
762 D. HEIMANN AND M. F. NEUTS
The first inequality follows by the induction hypothesis.
Inequality (32b) is proved by means of the following:
K K
G n ,L' i (i,j)= ]£ p„G» _i,Lj(miii (i + v, L' x ),j — 1) 3= ^ p„G n i,/.«(min (i + v, I/),./ — 1)
v= c=0
s* 2 p„G„ , lt j(min (i + v, L';)J1)
= G„,i.»(i, j).
The first inequality follows from the induction hypothesis and the second is a consequence of Theorem 5.
Theorem 7 now follows immediately from Theorem 6 and the definitions o{J3 n (Li) and B n (Li).
THEOREM 7: If L( =£ L'{, then for n 2* 0,
(33a) Bn(Li)^B n (L'{),
(33b) B n {L[)^B n m).
The inequalities in Theorem 7 are of course intuitive. As the bound L\ increases, it becomes more prob
able that the queue becomes empty before reaching the queue length L\. It also becomes less probable
that the queue empties out before time n + 1, because more customers may be allowed to join the queue.
5. THE PROBABILITY OF TERMINATION OF THE BUSY PERIOD
If the queue is unstable, i.e., p > 1, the duration of the busy period is infinite with positive prob
ability. Denoting by 6, the probability that the busy period ends in finite time, we note that 6 is given by
the smallest positive root of the equation
(34) d=R[P(6)].
The quantity 6 is readily computed by successive substitutions or by Newton's method. Whenever
p > 1, the probability 6 is first computed. In the alternate case, 6=1.
In general, the queues for which the traffic intensity p is close to one present computational diffi
culties, because of the long tail of the distribution of the busy period. In such cases, a large number of
the probabilities B n need to be computed to obtain a detailed picture of the busy period. In the sequel,
we shall use the term "nearcritical" for those cases where 1 — p «£ 0.1.
6. STOPPING CRITERIA
In any practical use, the recursive or iterative procedure must stop after a finite number of steps.
A major question is to determine when it is best to terminate. We shall now consider this problem for
both the recursive process (for the bounded queue) and the iterative process (for the unbounded queue).
6. 1 The Bounded Queue
We stop at a value N such that
SINGLE SERVER QUEUE IV 763
1 m
(35a) — V /8 w _ m+< (L 1 )<e,
m Mi
and
0ftv(L,)<8,
(35b)
where m, e, and 8 are given. Equation (35a) is the condition that an average of m consecutive density
values, ranging from fi N  m+ i(Li) to(3\(Li), be small, while (35b) requires that the cumulative probabil
ity be close to its limit 6. Clearly, the value of N will be higher in the case of nearcritical queues than
for other types of queues.
It should be pointed out that in actuality, the cumulative probability of the truncated unstable queue
converges to 1, rather than the 6 defined in (34). However, in the cases which we have examined, the
cumulative probability tends to stay close to 6 for many time points, and thereafter approaches one very
slowly. We may therefore use 6 as the critical value as discussed in the preceding paragraph. This is
plausible because the only way the busy period of the truncated queue can end earlier than that of the
unbounded queue is for it to empty out despite its length having reached the upper bound Li. The prob
ability of this happening within a reasonable time is very small, since the queue is inherently unstable.
6.2 The Unbounded Queue
In this case, the termination criterion is that the difference between two successive iterates for the
cumulative distribution function be very small. We therefore stop when:
(36) max \ k B m — k \B m \ < c,
l«msAf
where kB_ and kiB are the successive iterates and c and TV are given. The value of A; will again be higher
for nearcritical queues than for other types.
7. APPROXIMATION OF THE UNBOUNDED QUEUE
The busy period probabilities of the bounded queue may be used to approximate those of the
unbounded queue. The approximation is quite good, and it saves the rather large amount of computer
time required by the iterative process. We proceed to develop the approximation.
Let G ( n u) (i» j) be the probability than an unbounded queue becomes empty no later than time n.
The recurrence relations for G^(i, j) may be obtained from Equation (24) by letting L, * x. Clearly
(37) B n = 2rjGM(hJ),
J=i
where B„ is the cumulative distribution function of the busy period for the unbounded queue.
The following theorem is similar to Theorem 5. Its proof is therefore omitted.
764 D. HEIMANN AND M. F. NEUTS
THEOREM 8: For all i,, i 2 and./, such that, «£ i t «£ i 2 *£ In, and ()«£./*£ L 2 , we have
C»(»,»»C»(i,j).
for all n 3= 0.
The following theorems establish the approximation:
THEOREM 9: For any L[, L'{, and 0*£ i ^ min (Li, L?), 1 *£ j^L 2 , and n ^ 0, we have Gn,Li(i,j)
«sGM(*,j)^G».n(*.j).
PROOF: We consider the case where i 3» 1 andj > 1. The other cases are similarly proved.
The proof is by induction on n. The theorem is obviously true if n = 0. Assume it is true for n — 1.
Then
mtn(tf,LSil) _ K
G n ,L' 1 (i,j)= ^ P^i,«(* +"»/'— 1)*2 P"G*i>u(i+v,j— 1)
v=0
Sp^ 1 ('+"rl)=C ( ; ) (i,i),
v=0
where the last inequahty follows by the inductive hypothesis.
Furthermore,
W<» = £ pW\(i + v, jl) *s J p,GW (min (i + v, L'/),7D
i>=0 i»=0
*£ ^ p„G B i,LT(min (i + v> L"), j—l)=G n ,L" 1 (i, j).
The first inequahty is justified by Theorem 8, while the second follows from the inductive hypothesis.
THEOREM 10: For any L[, L", and for n 3* 0, the following inequahty holds:
PROOF: The theorem follows immediately from Theorem 9 and Equations (23), (29), and (37).
fin is therefore known to he within the interval [B n (Lt), B n (Li)], for any given L\. AsZi increases,
the interval length, which we denote by e„, becomes smaller, and thus a better approximation.
For queues which are not nearcritical, e„ is quite small, usually less than 10" 5 . This can be ex
plained by the fact that e n represents the probability that the queue empties out before time n, despite
its length reaching the upper bound L\ at least once before this occurrence. In the case of a stable
queue, the probability of ever reaching L\ is very small. For an unstable queue, while the probability
of reaching L\ is high, that of emptying out subsequently is very small.
However, the approximation is not as good for nearcritical queues. This is because both the above
probabilities are now significant.
8. COMPUTATIONAL ASPECTS OF THE UNBOUNDED QUEUE
Consider once again Equation (21) for the iterative process
SINGLE SERVER QUEUE IV 755
UK
i> =
This is more readily evaluated by use of Horner's method
(38) k p = a + k iP* (ai + kiP*(a 2 + . . . + fc _,/3 * (a^i + fc,/3 *a,, tK ) . . .)).
The advantage of (38) is that it requires only two arrays of length N (one for *£ and one for *i/3),
whereas (21) requires three (one for k (3, one for k  ,/J, and one for the intermediate storage of the arrays
k\P {v) as they are calculated and used in the equation).
We now take a count of the number of multiplications required to compute k /3. This is a reasonable
indicator of the computation time necessary for an algorithm, since multiplications take a great deal
more time than additions and so have the largest effect on the total computation time.
The first quantity computed is kifi*a L2K  Since (a/. 2 /f), = except for i = L 2 , the above quantity
needs N — L 2 multiplications for its calculation. Let y,. 2 Ki = a/, 2 *_i + fr i/3 * a,. lK , and y v = a„ + fc _i/3 * y v+u
v = 0, 1, . . ., LzK — 2, so thaty = fr/3. To compute y„ requires N(N—l)/2 multiplications, since
;i
{jv)i— (a v )i+ ^) (kifi)j(y v +i)ij requires i— 1 multiplications, for i = 2, 3, . . ., A^. Therefore, to
j=i
compute yo = fc/3, a total of
N(Nl)
(39) MC=_ ^ (L 2 Kl) + (/VL 2 )
multiplications are required.
We use the multiplications count (39) to ascertain the sensitivity of the computation time to N,
L 2 , and K. Since the first term of (39) dominates the second, the value of MC, and thus the computation
time, is approximately proportional to the second power of N, and the first power of L 2 and K. The
computation is therefore most sensitive to N.
To find an estimate for the total machine time required to run the program for any given value of
TV, we execute it twice for two relatively low values of N, and use the run times for them to find the
values a and 6 in the following equation derived from (39):
(40) T= aiN(Nl) + b,
where T is the estimated run time, i is the number of iterations executed by the program, a is a constant
of proportionality, and b is that part of the run time not spent in iteration, e.g., input and output, and
computing the initial estimate.
As an example, we consider an unstable queue with L 2 = 6 and K = 2. The two trial runs were car
ried out with A^= 100 and ^ = 250, and the run times were 4 and 11 seconds, respectively. Each run re
quired one iteration. We obtain
4 = 9900a + 6
11 = 62250a + 6,
766 D. HEIMANN AND M. F. NEUTS
which yields the solution, a = 0.0001337, 6 = 2.676 sec. Using these values, we calculate the following
estimates for the computation time:
N
Estimated
time
Actual time
100
4
4
250
11
11
500
36
37
650
59
63
These results give a good estimate of the maximum value of N which can be handled within a
given time limit.
Equation (40) is a special case of a general equation relating the estimated computation time to
N, L\, L 2 , K, and M (the number of time points for which the initial estimate is calculated):
(41) T=aiN(Nl)L 2 K+bML 1 L 2 K+c.
The first term of (41) represents the time used by the iterative process itself; the second term
represents the time for computing the initial estimate; and the third represents miscellaneous "house
keeping" time (which is usually quite small).
A technical report, containing program listings, numerical examples, and processing time data
is available from the authors upon request.
REFERENCES
[1] Dafermos, S. and M. F. Neuts, "A Single Server Queue in Discrete Time," Cahiers du Centre de
Recherche Operationnelle, 13, 2340 (1971).
[2] Klimko, E. and M. F. Neuts, "The Single Server Queue in Discrete TimeNumerical Analysis
II," in Naval Research Logistics Quarterly, 20, 297304 1973.
[3] Neuts, M. F.,"The Single Server Queue in Discrete TimeNumerical Analysis I," Naval Research
Logistics Quarterly, 20, 305319 1973.
[4] Takacs, L., Introduction to the Theory of Queues (Oxford University Press, New York, 1962).
[5] Titchmarsh, E. C, The Theory of Functions (Oxford University Press, 1939).
[6] Whittaker, E. T. and G. N. Watson, A Course of Modern Analysis (Cambridge University Press,
1927).
ALTERNATE METHODS OF PROJECT SCHEDULING
WITH LIMITED RESOURCES
James H. Patterson
College of Business Administration
The Pennsylvania State University
ABSTRACT
The applicability of critical path scheduling is limited by the inability of the algorithm
to cope with conflicting resource demands. This paper is an assessment of the effectiveness
of many of the heuristic extensions to the critical path method which resolve the conflicts
that develop between the resources demanded by an activity and those available. These
heuristic rules are evaluated on their ability to solve a large multiproject scheduling problem.
INTRODUCTION
PERT and CPM scheduling techniques implicitly assume that resources are available to schedule
activities at their technologically determined start times. Those who are familiar with these techniques
are aware, however, that activities often demand more resources than are available. When this happens,
a decision must be made regarding which activities to delay and which activities to schedule.
This constrainedresource problem has not yielded readily to the solution techniques of mathe
matical programming. Attempts to solve the resourceconstrained, project scheduling problem by
linear [27], integer [2], quadratic [1], and zeroone programming [21], as well as by bounded enumera
tion [4] and by branch and bound [10] have been successful only on small sets of moderate size projects.*
For the majority of practical project scheduling problems, the methods are inappropriate.
Because this problem is not easily solved by the available mathematical programming routines,
numerous efforts [15], [16], [27] have been expended in developing elaborate computerbased heuristic
solution procedures in order to determine feasible solutions to the problem. To date, the more com
putationally practical programs have been those employing heuristic methods of solution. In Wiest's
latest version of his SPAR2 program, for example, a problem with 1,500 jobs, 500 nodes, and prac
tically no limit on the number of resource categories or length of the project in days can be accom
modated [26].
While massive efforts have been expended in developing heuristic scheduling algorithms, very
little effort has been expended in measuring the relative effectiveness of the heuristics which they
employ. (Comparative studies which have appeared in the open literature are examined in the next
section). The present investigation is an attempt to assess statistically the scheduling ability of various
heuristic extensions to the critical path method in order to determine which ones are most likely to
develop improved schedules. Actual multiproject data are used in this assessment, and a full factorial
*Small sets of moderate size projects refer to problems containing no more than four projects of 30 or less activities each,
with each activity demanding less than five different resource types.
767
768 J H. PATTERSON
experiment is constructed to assess the effects of different project scheduling heuristics on various
measures of organizational performance to be described.
REVIEW OF COMPARATIVE STUDIES
One of the first attempts at describing the comparative performance of selected heuristic ex
tensions to the critical path method was performed by Brand, Meyer, and Shaffer in 1964 [2]. They
solved one multiresource test problem using computerized versions of Perk's "MSP" [20], Shirley's
"CPMS" [23], and Meyer and Shaffer's "RSM" [22]. They concluded that Perk's MSP was the best
of the three tested, giving preference to the information printed out by each method as well as to the
minimum duration schedule produced.
Fendley [7] in 1968 reported on a study performed to assess the effectiveness of selected project
scheduling heuristics in satisfying eight different measures of organizational performance. Eight mock
projects were formulated with each project having up to 20 activities and each activity requiring
0, 1, 2, or 3 units each of three scarce resources. Two and five project combinations of the eight hypo
thetical projects were then selected and scheduled with eight different heuristics (Shortest Imminent
Operation, Least Total Float, etc.). Two hundred iterations of the experiment were performed. Fendley
concluded that "the quantitative results did not unanimously confirm any one priority rule as best
for all resource availability levels in any performance category," although the Least Total Float rule
(minimumslackfirst) ranked first by four of the criteria selected and hence was judged most effective
overall.
Knight [11] examined the multiproject scheduling problem under a restrictive assumption that
did not allow for parallel activities in a project network. Knight's experimental medium was six sets
of 10 or 11 projects consisting of two to seven activities per project. Four different resource categories
were used, and each project set was scheduled using four different heuristics (Shortest Project Next,
Longest Project Next, Resource Utilization Ranking, etc.). Knight's objective was to minimize make
span, the time required to complete all projects. His general conclusion was that "rules based on
resource usage by each project are superior to those based on project length alone."
Mize [15] examined the job shop sequencing problem in a multiproject format. Eight hypothetical
multiproject organizations were created involving three, four, and six project sets. The largest organi
zation created involved 133 jobs (activities) and 20 departments. Under the assumptions that each
job could demand the services of only one department and that each department could be at work on
only one job at a time, Mize scheduled each organization using 12 different heuristics — three single
attribute heuristics and nine multipleattribute heuristics of his own devising. Using the objective of
"minimum project slippage," Mize concluded that three of the nine multipleattribute decision rules
generally yielded the best results of those tested. Each of these rules was some variation of the Least
Total Float heuristic.
Finally, Pascoe [18] was the first to suggest a classification scheme for project network param
eters (complexity, density, resource obstruction, etc.) and then to attempt to isolate the effect of these
parameters on the performance of project scheduling heuristics. His approach was to generate 32,
20activity networks with each activity requiring up to three resource types. Each of the generated
networks satisfied specified values of his project parameters. In total, 10 "common" heuristics (Least
Total Float, Late Finish Time, Late Start Time, etc.) were used to schedule each of the 32 projects.
Analysis of variance procedures (in the form of a factorial experiment) were used to analyze the re
sults. Unfortunately, the results of the experiment did not contradict a null hypothesis that there is
PROJECT SCHEDULING
769
no significant difference between results produced by different heuristics for the majority of the per
formance measures examined. Pascoe did conclude, however, that the heuristics of increasing Late
Finish Time or increasing Late Start Time produce the best results overall.
Thus there exists somewhat conflicting evidence as to the efficacy of many of the project scheduling
heuristics which resolve the conflicts that develop between the resources demanded by an activity
and those available. These inconsistencies are due partly to the nature of the data examined (the
majority of the data is hypothetical) and partly also to the selection of heuristics and objective functions
evaluated.
In the sections which follow, data describing an actual multiproject scheduling problem in a
research and development department of a U.S. Navy installation are presented, and attempts by the
installation to determine which scheduling rules have the highest probability of satisfying the measures
of organizational performance desired are described. Then, in order to generalize the results of this
investigation, scheduling problems appearing in the open literature are examined.
DESCRIPTION OF DATA
Data based on a scheduling problem at the Research and Development Department of the U.S.
Naval Ammunition Depot, Crane, Indiana, are used to test the scheduling rules evaluated. A network
diagram showing the sequence of activities of a typical project in this R&D organization is shown in
Figure 1.
Each project consists of approximately six distinct stages; the number of stages fluctuates because
of prior developmental work in a component item. Within each stage, many items are manufactured and
STAGE I
CONCEPT
DEVELOPMENT
STAGE 2
FEASIBILITY AND INITIAL
DEVELOPMENT EFFORT
STAGE 3
DETAILED PERFORMANCE AND
ENVIRONMENTAL TESTS 
REFINEMENT AND OPTIMIZATION
STAGE 4
ASSEMBLY AND
MANUFACTURE OF
EVALUATION LOT
STAGE 5
EVALU
ATION
PROGRAM
AND
TESTS
STAGE 6
DESIGN
REVIEW AND
RELEASE TO
PRODUCTION
LESS THAN 25 UNITS
200300 UNITS
300800 UNITS
RESOURCES AVAILABLE
TYPE
QUANTITY
ENGINEERS
21
CHEMISTS AND PHYSICISTS
14
TECHNICIANS
20
•MACHINISTS
8
ASSEMBLERS
30
TOOL DESIGNERS
6
PROCURERS
4
• PERFORMANCE EVALUATORS
10
ELECTRONIC EQUIPMENT DESIGNERS
6
• ADDITIONAL RESOURCES AVAILABLE
NON R8D
Figure 1. R&D Project flow.
770
J. H. PATTERSON
then tested. Depending upon the results of this testing, the entire stage is either repeated or the product
moves to the next stage for further developmental work. In the final stage, a report is issued either to
release the item for production, or to recommend that it not be produced.
Figures 2 and 3 and Tables 1 and 2 show the distributions of selected network and resource
characteristics for the problem considered. These 34 projects represent the work performed by this
organization over a 10month period, the length of time for which data are available. If a total project
cannot be technologically completed within a 10month interval, only the portion which can be com
pleted is considered.
The project characteristics shown in Figures 2 and 3 and Tables 1 and 2 differ somewhat from
those described in research efforts in which projects were generated on a computer [4], [10], [18].
For example, other researchers characteristically assume that three different resource types or groups
are the maximum that can be required to complete any one activity. From Figure 2, as many as 7 of
0.5
o" 0.4
—
UJ
ID
ti 0.3
rr
u.
0.2
0.1
1
1
12 3 4 5 6 7
CATEGORIES OF RESOURCES DEMANDED
FIGURE 2. Distribution of categories of different resources demanded by each activity (each activity could demand resources
from as many as 13 different resource categories).
0.75
0.77
0.79 OBI 0.83
0.85 0.87 089 091
COMPLEXITY
093
0.95 097 0.99
FIGURE 3. Distribution of project complexity (complexity is equal to the number of nodes, or jobs (N) in a project divided by
the number of arcs, or precedence relations (M). In the related job shop problem, the complexity is equal loNj{N— 1).
Table 1.
PROJECT SCHEDULING
Characteristics of Projects Scheduled
771
Project
Total mandays
Critical path length
number
required
(in days)
1
1,125
136
2
728
105
3
639
90
4
365
54
5
153
91
6
89
54
7
317
101
8
447
99
9
129
93
10
393
90
11
393
96
12
111
72
13
297
45
14
512
81
15
255
70
16
2,278
153
17
280
70
18
355
81
19
356
104
20
113
55
21
109
46
22
402
76
23
274
52
24
527
96
25
201
42
26
2,315
159
27
341
97
28
382
113
29
965
105
30
382
156
31
259
80
32
151
65
33
717
134
34
462
103
13 possible resource groups can be required to accomplish an activity. Also, the size of the projects
scheduled varies substantially in mandays required to complete each project, while other simulation
studies have assumed fairly "constant" project size (as measured in resource requirements). Tables
1 and 2 and Figures 1,2, and 3 are included because field data suitable for comparing heuristic sched
uling methods are generally not available, either because they are proprietary or because resource
constraints are not explicitly considered and conventional CPM techniques are used.
SCHEDULING RULES EXAMINED
A variety of heuristic scheduling rules for multiproject problems can be found in the literature.
The studies of Fendley, Knight, Mize, Pascoe, etc. give only a sampUng of the number available. In
general, the origin of the scheduling heuristics in use lies in previous research efforts in the job shop
scheduling area and in the results of applications of the definitions indigenous to the critical path
method, such as Late Start Time, or Late Finish Time, etc. An attempt was made to categorize the
available heuristic scheduling rules so that samples from each classification could be selected and
772
J. H. PATTERSON
Table 2. Resource Statistics for Projects Scheduled
Re
Total available
Approximate
Average number of resource
source
Total required
direct project
resource
units demanded when
num
manhours
manhours
utilization a
required by an activity
ber
1
979
1,260
0.78
1.65
2
2,460
3,060
0.80
2.28
3
2,918
3,600
0.81
, 2.72
4
699
1,080
0.65
1.28
5
2,456
3,060
0.80
1.67
6
761
1,080
0.70
1.49
7
322
1,260
0.26
1.03
8
298
720
0.41
1.18
9
3,917
5,040
0.78
1.48
10
1,179
1,620
0.73
1.18
11
286
540
0.53
1.11
12
273
540
0.51
1.54
13
123
360
0.32
1.57
a Because of the method used to obtain replications of the experiment (by timephasing each of the individual projects),
the actual resource utilization will differ from these figures by a small amount.
applied to the multiproject scheduling problem, the classification scheme giving some assurance that
the population of available heuristics had been adequately sampled. These attempts led to the follow
ing conclusions:
1. Numerous project scheduling rules are based upon a welldefined objective, such as minimizing
the completion time of the projects or maximizing the utilization of employed resources.
2. Other scheduling rules are not related specifically to an objective function, but rather reflect
a characteristic of the project set or the activities comprising the project set. For example, longest
project next assigns resources to the activity of the project having the longest expected span; any
remaining activities are scheduled only if resources permit.
3. Still other scheduling rules do not relate to a specific objective function or to a characteristic
of a project set, but rather use other scheduling rules as a basis on which to build. Switch activities
at random in the order in which they are considered for resource assignment when they tie on another
priority is an example of such a scheduling rule.
4. Finally, a few scheduling rules base their rationale on learning from knowledge gained in
previous scheduling efforts, and are applied only after another scheduling rule has been used. Re
schedule based on resource conflicts exhibited in the first feasible schedule is an example of such a rule.
Based on the above classification and on preliminary computational experience, the heuristic
scheduling rules listed below were selected for evaluation. All rules are applied in conjunction with the
parallel method of scheduling in which activity priority is determined during scheduling rather than
before. All heuristic rules considered refer to the choosing of one activity over another for resource
assignment because of a difference in a priority or other index associated with that activity.
1. Least Total Float (LTF)
Schedule the activities on the basis of total float, those activities possessing the least total float
being considered first. The total float present in an activity is the difference between the late and early
start times determined through conventional critical path analysis.
PROJECT SCHEDULING 773
2. Greatest Total Resource Demand (CTRD)
Schedule the activities on the basis of the total resources required by the activity, that activity
with the greatest resource usage being scheduled first. The rationale behind this rule lies in scheduling
potential bottleneck activities because of the high usage of varied resources the activity consumes.
3. Greatest Remaining Resource Demand (GRRD)
Schedule the activities on the basis of the total remaining manhours (resources) of work required
on the project, those with the greatest remaining manhours being scheduled first. This rule is related
to many of the look ahead rules receiving attention in the literature today. It attempts to look ahead
further than the current activity in locating potential bottlenecks; it is an extension of GTRD and
changes values dynamically throughout the development of a schedule.
4. Shortest Imminent Operation (SIO)
Schedule the activities on the basis of the duration of the activity, that activity with the shortest
duration being scheduled first. This is analogous to the SIO rule of the job shop.
5. Greatest Resource Utilization (GRU)
Schedule the activities at time t to maximize the utilization of available manpower (resources).
This rule is formulated and solved as a zeroone integer programming problem where xj= 1 if activity
j(aj) is to be scheduled during the current time interval, and if it is not:
max ^ c j x j
i
subject to R'x^ bj
Xj€{0, 1}
b t= resource i available at time t; b* dimension q X 1
Q
c i= X rij
i=l
rtj— resource i required by ctj
/?'= matrix of resources required by all aj which can be scheduled at time t.
Note that r« can be defined for manpower, machines, etc., but that when different types of resources
are scheduled simultaneously, it becomes necessary to define common units for them. For example,
if both men and machines are to be scheduled at time t, a value (cost) can be associated with each,
in which case one maximizes the "value" of employed resources. For the data described, r y refers to
manpower, but ordinarily would not have to be restricted to this type of resource.
The application of the Greatest Resource Utilization rule demonstrates that heuristic scheduling
rules are not always based on priority indices for activities and subsequently cannot be implemented
solely through sorting routines.
774
J. H. PATTERSON
6. Randomness (RAN)
If any activities which are candidates for scheduling during the current time interval tie on any
given priority, switch them at random in the order in which they are considered for resource assign
ment. The randomness rule is never applied by itself. Despite the simplicity of this rule, attempts
[26] to solve certain project scheduling problems without it have been unsuccessful.
7. Reschedule (RES)
Determine which class or category of resource had the least total amount of idle time in the first
feasible schedule. Develop a new schedule using as activity priority the quantity of this resource
required, multiplied by the duration of the activity.
Other research efforts [11], [15] have determined that improvements can be made in schedules
either by rescheduling on the basis of a resource that caused more delays in activities in the first sched
ule, or by including provisions for gapclosing by rescheduling those activities that are scheduled to
start before their technological (CPM) late start time. The reschedule feature used here is representa
tive of the former reschedule routine.
Table 3. Scheduling Rules Examined
Scheduling rule
Identification
Basis for inclusion
Least total float
LTF
GTRD
GRRD
SIO
GRU
RAN
RES
Take advantage of the slack time present in an activity.
Schedule potential bottleneck activities.
Look ahead further than the current activity in scheduling
potential bottleneck activities.
Process as many jobs through the system as rapidly as possible
in an attempt to minimize delays.
Schedule all resources that can be scheduled during a time
interval.
Add an element of chance in the assignment of resources to an
activity.
Reschedule activities on the basis of critical or tight resources
exhibited in the first feasible schedule.
Greatest total resource demand
Greatest remaining resource demand
Shortest imminent operation
Greatest resource usage
Randomness
Reschedule
PERFORMANCE CRITERIA
The following criterion functions are used to assess the efficacy of the scheduling rules and sched
uling programs (combinations of scheduling rules) examined. The choice of an appropriate objective
function may differ in various scheduling environments and in different periods of time. Several of the
common ones are therefore selected for examination.
1. Total Project Delay
The total delay of a project set is the sum (over all projects) of the difference between the assigned
scheduled finish time of a project and the length of the critical path in an early start schedule. It is
doubtful that many of the projects considered in a resourceconstrained, multiproject scheduling
problem will be completed within the critical path completion time estimate. This measure does,
however, give an indication of the delays introduced as a result of limitations on resource availability
and as a result of the scheduling rule employed.
PROJECT SCHEDULING 775
2. Weighted Total Delay
The weighted total delay of a project set is the sum (again, over all projects) of the total resources
demanded by a project multiplied by the total delay of the project as defined above. Weighted delay
is measured in mandays for the data described. This measure places additional emphasis on the
seriousness of a delay in the larger projects.
3. Total Resource Idle Time
The total resource idle time is the amount of time that resources are idle during a schedule span.
Idle time is measured in manhours; it is a result of the unavailability of direct project work, which
in turn is a result of the scheduling method employed.
In order to avoid biasing this summary measure by using different time intervals for measuring
idleness, a measure of the idle resources up to and including 140 days after the start date for schedul
ing (day 0) was used. The limit of 140 days for accumulating idle resources represents approximately
threefourths of the length of the schedule span generated by each heuristic scheduling procedure.
4. Computer Processing Time
The computer processing time is the amount of time expended in generating a schedule. It is meas
ured in seconds and is an indication of the direct cost of using the scheduling rule chosen.
EXPERIMENTAL DESIGN
A randomized complete block, full factorial design is used to assess the efficacy of the heuristic
scheduling rules examined. The blocking performed is achieved by varying the starting times of each
of the 34 projects. These starting times are generated by drawing random variates which are uniformly
distributed over a span of 180 days, the length of a schedule. For each block, a string of 34 random
variates is drawn. The first random variate is the starting time of the first project, the second is the
starting time of the second project, etc. One string of 34 random variates corresponds to one block
of the experiment. This method of obtaining replications of the experiment has the effect of varying
the total resources required during the schedule span, and hence affects the resource usage rate.
Of the seven scheduling rules examined, five (LTF, GTRD, GRRD, SIO, GRU) are applied to a
project scheduling problem independently of one another. These five main scheduling rules are called
treatment A in the discussion which follows.
The remaining two scheduling rules (RAN & RES) are applied only in conjunction with one of
the first five rules. They are labeled treatments B and C, respectively. These last two treatments
exist at one of two possible levels; they are either present in a heuristic program or they are not. (This
is to be distinguished from treatment A, where one level of the factor must always be present in order
to develop a schedule.)
Thus, one factor (treatment A) is varied over five levels and each of two factors (treatments B
and C) are varied over two levels. The experiment is then a 5 X 2 X 2 factorial experiment; from seven
scheduling heuristics, 20 (5 X 2 X 2) treatment combinations are formed. All 20 treatment combinations
are investigated.
For each of the four descriptive measures evaluated, a fourway and a twoway analysis of variance
is given in order to assess interactions, main effects, and effects due to blocking or introducing different
resource demands. Duncan's Multiple Range Test [6] is then used to rank the various means; the
5percent level of significance is used for reporting significant differences.
776
J. H. PATTERSON
A total of 30 replications (blocks) of the experiment are made. This number is based on previous
multiproject scheduling research [15] and the desire to reject the hypothesis with the power at 0.95
and ot= 0.05 that the scheduling rules are equal when in fact one of them exceeds the others by cr, the
population standard deviation. Examination of power curves of the NonCentral FDistribution de
veloped by Pearson and Hartley [8], [19] reveals that a sample of n = 30 is sufficient to insure the
accuracy stated.
TEST RESULTS
A computer program called MPSP, an acronym for MultiProject Scheduling Program,* was
written to implement the scheduling rules described. The program was run on the Indiana University
CDC 3600 computer. The computer processing times herein reported refer to average time spent in
generating a schedule, exclusive of all 1/0 time.
Bartlett's [5], Cochran's [5], and Hartley's ShortCut [9] test for the homogeneity of variances
are made for all of the summary measures. With Bartlett's test, the hypothesis that the variances
associated with total project delay and computer processing time are homogeneous is rejected at the
5percent level of significance; the hypothesis for the measures weighted total delay and total resource
idle time is accepted. With Cochran's and Hartley's tests, the variances associated with each summary
measure are concluded to be homogeneous at the 5percent level of significance. The variances are
therefore treated as being homogeneous. These results are summarized in Table 4.
Having concluded that the variances are homogeneous, the statistical results of ranking each of the
scheduling rules are now given. t An attempt is also made to state which scheduling rule is superior
for each of the criterion functions examined.
1. Total Project Delay
The average amount of total project delay varies between 998 and 1,502 days depending upon the
scheduling rule used. The application of the Shortest Imminent Operation heuristic produces the least
total project delay; the heuristic Greatest Remaining Resource Demand produces the highest amount
of delay.
Table 4. Tests for Homogeneity of Variances
Criterion function
Bartlett's test
computed value
Cochran's test
computed value
Hartley's test
computed value
Total project delay
a 1.69
0.089
3.82
Weighted total delay
1.01
0.082
3.07
Total resource idle time
0.98
0.092
3.78
Computer processing time
a 1.81
0.091
3.65
a The variances are concluded to be homogeneous at the 5percent significance level with two exceptions. The hypothesis
of homogeneous variances is rejected at the 5percent level using Bartlett's test for the measures of total project delay and
computer processing time. The hypothesis of homogeneous variances is accepted using each test and for all of the summary
measures at the 1 percent significance level.
*MPSP was written by J. A. Werne of NAD Crane.
tThe results herein reported are coded, since they are not intended for public use. This coding does not affect the analysis
or the interpretation of the results.
PROJECT SCHEDULING
777
The presence of Randomness (Treatment B in Table 5) when used in conjunction with the other
scheduling heuristics is not generally expected to produce better results; it might on the average even
be disfunctional. But occasionally, the presence of some element of chance in the selection of activities
for resource assignment may lead to optimal solutions to project scheduling problems. Thus, as might
be expected, randomness contributes little in explaining the variation present in total project delay
as shown in Table 5.
It could conceivably be argued that the statistical analysis is not particularly appropriate when
the factor (treatment) for randomness is included in the analysis. However, for completeness, it is
included. If randomness were significant in the analysis of variance, this would demonstrate that the
heuristics being examined did not discriminate amongst the activities. For example, if all of the dura
tions of the activities were equal, the Shortest Imminent Operation heuristic would in essence be a
"Select Activities at Random" rule. In this instance, the randomness rule could be construed to be
another heuristic which should be included in Treatment A. Thus there is some information in the fact
that Treatment B is insignificant, albeit minor.
The main scheduling rule and the reschedule routine contribute significantly in explaining the
variation present in a schedule. The presence of rescheduling, however, generally increases the amount
of total project delay present. Only for the rules Greatest Remaining Resource Demand and Greatest
Total Resource Demand is total project delay reduced by the presence of the reschedule rule. Both of
these rules, however, represent the worst instances of total project delay.
Table 5. Analysis of Variance for Total Project Delay
Source
d.f.
S.S.
M.S.
F
^o=0.01
A
4
1
1
4
4
1
4
29
551
6,491,757
6,767
279,288
10,318
4,394,954
5,287
6,455
1,761,246
2,314,408
1,622,939
6,767
279,288
2,580
1,098,738
5,287
1,614
60,732
4,200
386.38
1.61
66.49
0.61
261.58
1.26
0.38
14.46
3.35
6.68
6.68
3.35
3.35
6.68
3.35
1.75
B
C
AB
AC
BC
ABC
Blocks
Error
Total
559
15,270,481
Table 6. Analysis of Variance for Total Project Delay
Main Scheduling Heuristic Only
Source
d.f.
S.S.
M.S.
F
^a=0.01
Treatments
Blocks
4
29
116
5,126,712
518,316
504,627
1,281,678
17,873
4,350
294.62
4.11
3.48
1.86
Error
Total
149
6,149,655
The results of applying Duncan's test to the observed data are shown in Table 7. The Shortest
Imminent Operation heuristic produces the lowest ranking sample mean.
778
J. H. PATTERSON
Table 7. Multiple Range Test for Total Project Delay
Mean
(in days)
£jr rtCM s £ir' rt 1/5 •<* .h © ooeaH pi © rf eo ©oo
©00 l/i tJ> © © \0 IrtlOlrtin rococo CM i— l Irt ** ©on
l/J ^ "JCT. ^i*^^. ""I "1 'I ""l nHrtrt rt rt © © ©On
Treatment
crt
Crt
£ 3 1
Q
OS
OS
O
Q
OS
H
O
O
OS
H
O
Q
OS
o
Q
OS
OS
o
Q
OS
H
O
tfi
3
OS
o
W
OS
z
<
as
OS
O
en
w
OS
z
OS
u
OS
[5
w
OS
Q
OS
5
w
OS
3
z
1
z
1
H
1
3
as
o
OS
O
15
o
35
o
A horizontal line enclosing a group of means indicates that the means located within the group cannot be distinguished from
one another at the 5percent level of significance. Descriptions of each of these treatments are found in Table 3.
2. Weighted Total Delay
The scheduling heuristic which produces the least weighted total delay is Least Total Float.
Duncan's multiple range test is, however, unable to distinguish between this scheduling rule and the
Greatest Resource Usage rule. Since the Least Total Float rule requires approximately onehalf of the
computer processing time that the Greatest Resource Usage rule requires, Least Total Float is the
better scheduling rule to use to minimize weighted total delay. Results of applying Duncan's test are
shown in Table 10.
Table 8. Analysis of Variance for Weighted Total Delay
Source
d.f.
S.S.
(100,000)
M.S.
(100,000)
F
F a=0.01
A
B
C
AB
AC
BC
ABC
Blocks
Error
Total
4
1
1
4
4
1
4
29
551
192,859
1,699
19,963
4,022
130,052
953
3,218
417,217
806,356
48,215
1,699
19,963
1,005
32,513
953
805
14,387
1,463
32.95
1.16
13.64
0.69
22.22
0.65
0.55
9.83
3.35
6.68
6.68
3.35
3.35
6.68
3.35
1.75
599
1 ,223,573
Table 9. Analysis of Variance for Weighted Total Delay
Main Scheduling Heuristic Only
Source
d.f.
S.S.
(100,000)
M.S.
(100,000)
F
F a =0.0\
Treatments
Error
Total
4
29
116
180,315
117,411
232,388
45,079
4,049
2,003
22.50
2.02
3.48
1.86
149
530,114
PROJECT SCHEDULING
Table 10. Multiple Range Test for Weighted Total Delay
779
Mean
(in mandays)
to m r cm
» Oi M »
(C (^ rt *_
CO CO fO H
CM CM CM CM
t— r t— r
s
rH © r^ r^ i/5 © r~
S * £■ t w s N
®i © "V ^1 6r "i "i
■* to' ©' to' to CM 00
rH pH ^i © © © o
r~ r^ t^ o» t> r~ \©
© O £ CM
O CM t— "5
CM. J ^ C *i
t— m' to cm
O O Ctn O
vO \C 'C vO
660,190
655,869
653,746
644,648
Treatment
GRRD RAN RES
GTRD
GRRD RES
SIO
z
a
5
OS
o
GTRD RAN
GRRD
SIO RAN
LTF RAN RES
GRU RAN RES
GTRD RAN RES
SIO RAN RES
GRU RES
SIO RES
GTRD RES
LTF RES
GRU RAN
LTF RAN
GRU
LTF
A horizontal line enclosing a group of means indicates that the means located within the group cannot be distinguished
from one another at the 5percent level of significance. Descriptions of each of these treatments are found in Table 3.
The four way analysis of variance given in Table 8 shows treatment A, treatment C (the reschedule
feature), and the A~C treatment interaction are significant beyond the 1percent level of significance in
the analysis of variance.
3. Total Resource Idle Time
The scheduling abilities of the Shortest Imminent Operation heuristic and the Greatest Remaining
Resource Demand rule are reversed when measuring total resource idle time as opposed to total project
delay. The Greatest Remaining Resource Demand rule produces schedules with the least total resource
idle time, and the Shortest Imminent Operation heuristic produces schedules with the greatest amout
of resource idle time. Such a result can be expected. The logic behind the Greatest Remaining Resource
Demand rule lies in scheduling potential bottleneck activities and hence in utilizing resources efficiently,
while the logic of the Shortest Imminent Operation heuristic lies in accomplishing as many jobs as
possible in as short a length of time as possible. The results of applying Duncan's test to the criterion
total resource idle time are shown in Table 13.
The fourway analysis of variance for total resource idle time is shown in Table 11. As with the
previous measures for assessing scheduling ability, treatments A and C and the AC treatment inter
action are highly significant in explaining the variation present in schedules, whereas treatment B,
randomness, is insignificant.
Table 11. Analysis of Variance for Total Resource Idle Time
Source
d.f.
S.S.
M.S.
F
Fa=0M
A
4
1
1
4
4
1
4
29
551
971,843,896
168,371
579,515,883
5,948,609
639,991,514
12,595
8,764,936
1,011,357,775
1,697,964,539
242,960,974
168,371
579,515,883
1,487,152
159,997,879
12,595
2,191,234
34,874,405
3,081,605
78.84
0.05
88.06
0.48
51.92
0.00
0.71
11.32
3.35
6.68
6.68
3.35
3.35
6.68
3.35
1.75
B
C
AB
AC
BC
ABC
Blocks
Error
Total
599
4,915,568,097
780
J. H. PATTERSON
Table 12. Analysis of Variance for Total Resource Idle Time
Main Scheduling Heuristic Only
Source
d.f.
S.S.
A/.S.
F
F a = 0.0l
Treatments
4
29
116
790,870,250
274,677,502
387,737,532
197,717,563
9,471,638
3,342,565
59.15
2.83
3.48
1.86
Blocks
Error
Total
149
1,453,285,284
TABLE 13. Multiple Range Test for Total Resource Idle Time
Mean
s
ffl
r
1/3
O
3
Ov
ov
CO
co
8!
1—
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IS
00
to
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?
CO
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"l
"i
31
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"1
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00
v©
8
S
$
«
8
s
58
$
m
vO
3
CO
vo
CO
CO
vO
o
co
co
co
CO
co
z
co
co
Z
Z
Z
Treatment
Z
<
as
z
2
z
CO
3
Q
2
Q
co
Z
2
Q
Z
2
1
Q
a
Q
Z
2
a
Q
O
O
3
as
O
ft
as
H
OS
OS
P
O
ft
OS
OS
ft
OS
OS
ft
as
P
OS
p
3
OS
3
OS
OS
OS
OS
OS
co
CO
O
CO
J
O
o
O
co
J
o
J
O
J
O
o
o
o
o
o
A horizontal line enclosing a group of means indicates that the means located within the group cannot be distinguished
from one another at the 5percent level of significance. Descriptions of each of these treatments are found in Table 3.
4. Computer Processing Time
The computer processing time criterion is one measure of the cost of operating a scheduling
system. Table 16 shows the average time (in seconds) required to develop a schedule. As can be seen,
the six lowest ranking sample means differ at the most by only 3 sec (CDC 3600 CPU time). And while
the sixteen lowest ranking sample means differ by as much as 46 sec, this difference is of little economic
significance. Computer processing time is therefore concluded to be of little economic significance
in developing heuristic schedules for the scheduling problem considered.
Table 14. Analysis of Variance for Computer Processing Time
Source
d.f.
S.S.
M.S.
F
F a =0.0l
A
4
1
1
4
4
1
4
29
551
608,183
648
54,037
394
154
30
2,176
10,185
152,046
648
54,037
98
39
30
75
18
8225.61
35.08
2923.39
5.33
2.09
1.65
4.06
3.35
6.68
6.68
3.35
3.35
6.68
3.35
1.75
B
C
AB
AC
BC
ABC
Blocks
Error
Total
599
675,808
PROJECT SCHEDULING
TABLE 15. Analysis of Variance for Computer Processing Time
Main Scheduling Heuristic Only
781
Source
d.f.
S.S.
M.S.
F
^"0=0.01
Treatments
Blocks
4
29
116
146,386
811
3,040
36,596
28
26
1396.65
1.07
3.48
1.86
Error
Total
149
150,237
Table 16.
Multiple Range Test for
Computer
Processing Time
Mean
CM
CM
r cm no
00 v©
irt 00 r
3! ea
CM Q
© 00 W CM
* «5 CM CM
CM
CM
CM
CM CM
(in seconds^
^H
iH 1— 1
3
w
OS
C/5
Z en
Z crt
« Z Z
Treatment
<
OS
t/5
W
OS
5 z
23 1
t/3 Z
£ ol
OS OS
a q
3 w
OS oS
Q Q
OS 5 Q Q
Q
a
z
3
OS
OS
3 3 U.
OS OS H
& £ &
OS OS
OS OS
OS OS
2 2 h 1
OS
H
OS
OS
OO J
J J J
O O
c/3 c/i O O
O
en crt
A horizontal line enclosing a group of means indicates that the means located within the group cannot be distinguished
from one another at the 5percent level of significance. Descriptions of each of these treatments are found in Table 3.
PROJECT SCHEDULING EXPERIENCE: HYPOTHETICAL DATA
Six sets of fictitious projects ranging in size from four to six projects each were adapted from
[15] and scheduled with MPSP. These data differ substantially from that previously analyzed in that
each activity requires only one resource type, and each of the 20 available resource types can be at
work on no more than one activity at a time. This then is actually job shop sequencing data viewed in
a multiproject format. The analysis of this data presents a different type of challenge to the scheduling
rules than does the data previously examined.
These project sets were scheduled with each of the six sets being considered a different replicate
of the experiment. Table 17 presents the analysis of variance results for the criterion total project delay.
The notation is the same as used in the previous section, with treatment A being the main scheduling
heuristics, treatment B being randomness, etc.
The FRatio for replications shown in Table 17 is far greater (414.8 > 10.5) than theFRatio for the
main scheduling heuristics. This indicates that for the hypothetical data considered, project and re
source characteristics contribute more in explaining the variation present in project schedules than
do the scheduling methods employed.
Analysis of variance results for each of the remaining summary measures produced similar results
and hence are not shown; the FRatio for replicates is easily 20 times as large as the FRatio for any
of the remaining sources of variation. No reason is given in the reference as to why the particular
structure was chosen for the project sets producing these results, but results similar to the above have
been noted when analyzing the single project problem using hypothetical data [18].
782 J. H. PATTERSON
Table 17. Analysis of Variance for Total Project Delay: Hypothetical Data
Source
d.f.
S.S.
M.S.
F
A
4
1
1
4
4
1
4
5
95
72,141
20
29,954
333
53,367
469
54
3,556,728
162,902
18,035
20
29,954
83
13,341
469
14
711,345
1,715
10.52
0.01
17.47
0.05
7.78
0.27
0.01
414.84
B
C
AB
AC
BC
ABC
Replications
Error
Total
119
Since the FRatio for replicates is so large, no attempt was made to rank the scheduling heuristics
using multiple ranking procedures. The sample means for the main scheduling heuristics are, however,
shown in Table 18 omitting the statistical analysis. The ordinal ranking of many of the heuristic schedul
ing rules is similar to that noted in examining real data.
Table 18. Sample Means for Main Scheduling Heuristic: Hypothetical Data
Scheduling heuristic
Performance criteria
Total project
delay a
Total weighted
delay a
Total resource
idle time a
Computer processing
time b (sec)
Least total float ,
421
482
484
j 438
460
260,155
302,830
279,857
275,166
286,158
4,802
5,097
4,447
5,007
4,892
3.59
1.95
2.13
1.99
7.02
Greatest total resource demand
Greatest remaining resource demand
Shortest imminent operation
Greatest resource usage
a Since hypothetical data are used, no units are attached to these means.
" CDC 3600 CPU Time.
An attempt was also made to schedule a single project problem posed by Martino [14]. This
problem was fabricated to produce bad results using many of the heuristic scheduling rules available.
With a resource limit of eight men and without using the random switching feature, MPSP is unable to
produce an optimal* schedule for this problem. With the random switching feature turned on, however,
the algorithm found an optimal schedule using the Greatest Resource Usage heuristic. The algorithm
was also able to identify five different optimal schedules when it was allowed to repeat the Greatest
Resource Usage rule with randomness a total of 20 times. These results are similar to those reported
by Wiest [26].
CONCLUSION
Optimal solutions to the constrainedresource, project scheduling problem are infeasible at the
present for all but moderate size problems. And while heuristic approaches offer the most promising
results to date, several heuristic approaches should be examined and compared to the current require
*The critical path length for this project is 19 days and the project can be accomplished with as few as eight men in a 19day
period. Because the project can be completed in nineteen days with no variation in resource usage and no idle resource time,
a 19day schedule is considered optimal
PROJECT SCHEDULING 783
merits of a good schedule before any one of them is selected. The results reported demonstrate that
it is economically practical to employ several heuristic scheduling rules and to then choose a schedule
which comes closest to meeting desired objectives.
The results of examining actual and laboratory type projects suggest that the scheduling rules
which produce schedules with low total project delays do so at the expense of an inefficient utilization
of resources; the scheduling methods which schedule resources efficiently do so at the expense of
large delays in project completions. While it was not possible to distinguish among heuristic scheduling
rules when hypothetical data were examined, the results reported demonstrate that when actual project
data are considered, the scheduling methods employed account for a significant portion of the variation
present in project schedules.
It has been suggested that the evaluation of heuristic models should be based on their usefulness
rather than on their ability to obtain optimum solutions to problems. While the MultiProject Scheduling
program was able to identify five optimum solutions to a specific problem, in general it cannot be
claimed that an optimum is obtainable by these methods.
Field data suitable for comparing heuristic scheduling methods are generally not available either
because they are proprietary or because resource constraints are not explicitly considered and conven
tional CPM techniques are used. The inclusion of an actual multiproject, constrainedresource schedul
ing problem should aid researchers in the area in comparing other heuristic scheduling algorithms and
in selecting characteristics for generating laboratory projects.
BIBLIOGRAPHY
[1] Bennett, Fred L., "Some Approaches to the Critical Path Scheduling Resource Allocation Prob
lem," unpublished Ph.D. Thesis, Cornell University (1966).
[2] Brand, J. D., W. L. Meyer, and L. R. Schaeffer, "The Resource Scheduling Problem in Con
struction," Civil Engineering Studies, Report No. 5, Urbana, Illinois: Department of Civil
Engineering, University of Illinois (1964).
[3] Davies, Owen L. (ed.), The Design and Analysis of Industrial Experiments (Oliver and Boyd,
London, 1960).
[4] Davis, E. W., "An Exact Algorithm for the Multiple ConstrainedResource Project Scheduling
Problem," unpublished Ph.D. Thesis, Yale University (1969).
[5] Dixon, Wilfrid J. and Frank J. Massey, Jr., Introduction to Statistical Analysis (McGrawHill
Book Company, Inc., New York, 1969).
[6] Duncan, Acheson J., Quality Control and Industrial Statistics (Rev. Ed.) (Richard D. Irwin, Inc.,
Homewood, Illinois, 1959).
[7] Fendley, L. G., "Toward the Development of a Complete Multiproject Scheduling System," The
Journal of Industrial Engineering, Vol. 19, No. 10 (Oct. 1968), pp. 505515.
[8] Guenther, William C, Analysis of Variance. (PrenticeHall, Inc., Englewood Cliffs, New Jersey,
1964).
[9] Hartley, H. O., "The Maximum FRatio as a ShortCut Test for Heterogeneity of Variance,"
Biometrica, Vol. 37 (1950), pp. 308312.
[10] Johnson, Thomas J. R., An Algorithm for the ResourceConstrained, Project Scheduling Problem.
Unpublished Ph.D. Thesis, Massachusetts Institute of Technology (1967).
784 J. H. PATTERSON
[11] Knight, R. M., "Resource Allocation and MultiProject Scheduling in a Research and Develop
ment Environment," Unpublished M.S. Thesis, Massachusetts Institute of Technology (June
1966).
[12] Lambourn, S., "RAMPS — A New Tool in Planning and Control," The Computer Journal, Vol. 5
(1963), pp. 300304.
[13] Levy, F. K., G. L. Thompson, and J. D. Wiest, "MultiShip, MultiShop Workload — Smoothing
Programs," Nav. Res. Log. Quart. 9, 3745 (Mar. 1962).
[14] Martino, R. L., Project Management and Control: Vol. Ill, Allocating and Scheduling Resources,
(American Management Association, New York, 1965).
[15] Mize,J. H., "A Heuristic Scheduling Model for MultiProject Organizations," unpublished Ph.D.
Thesis, Purdue University (1964).
[16] Moshman, J., J. Johnson, and M. Larsen, "RAMPS — A Technique for Resource Allocation and
Multiproject Scheduling," Proceedings — 1963 Spring Joint Computer Conference, pp. 1727.
[17] Muth, John F. and Gerald L. Thompson (Eds.), Industrial Scheduling (PrenticeHall, Inc., New
Jersey, 1963).
[18] Pascoe, T. L., "An Experimental Comparison of Heuristic Methods for Allocating Resources,"
unpublished Ph.D. Thesis, Cambridge University (1965).
[19] Pearson, E. S. and H. O. Hartley, "Charts of the Power Function for Analysis of Variance Tests
Derived from the NonCentral FDistribution," Biometrica, Vol. 38 (1951), pp. 112130.
[20] Perk, H. N., "ManScheduling Program for the IBM 1620," Revised, IBM Program Library File
No. 10.3.013.
[21] Pritsker, A. A. B. and L. J. Watters, "A ZeroOne Programming Approach to Scheduling with
Limited Resources," The RAND Corporation, RM5561PR (Jan. 1968).
[22] Shaffer, L. R., J. B. Ritter, and W. L. Meyer, The Critical Path Method (McGrawHill, New York
1965).
[23] Shirley, W. W. and I. L. Bowman, "8,000, Critical Path and Man Scheduling," Richfield Oil Co.
Los Angeles, California (1962).
[24] Tonge, Fred, "The Use of Heuristic Programming in Management Science," Management Science
Vol. 7, No. 1 (Oct. 1960), pp. 2142.
[25] Wiest, J. D., "Some Properties of Schedules for Large Projects with Limited Resources," Opera
tions Research, Vol. 12, No. 3 (MayJune 1964), pp. 395418.
[26] Wiest, J. D., "A Heuristic Model for Scheduling Large Projects with Limited Resources," Manage
ment Science, Vol. 13, No. 6 (Feb. 1967), pp. 359377.
[27] Wiest, J. D., "The Scheduling of Large Projects with Limited Resources," unpublished Ph.D
Thesis, Carnegie Institute of Technology (1963).
[28] Winer, B. J., Statistical Principles in Experimental Design. (McGrawHill Book Company, Inc.
New York, 1962).
OPTIMUM ADJUSTMENT POLICY FOR A PRODUCT WITH
TWO QUALITY CHARACTERISTICS
William J. Kennedy, Jr.
University of Utah
and
Prabakhar M. Ghare
Virginia Polytechnic Institute and State University
INTRODUCTION
We consider the problem of determining an optimal adjustment policy when the price received
for the product is a function of a stated quality measure. When this quality measure has a specified
value maximum price can be received. As the quality measure deviates from the specified value the
price received drops progressively. An example of this type of pricing is shown in Table 1.
The quality measure is a function of two characteristics X and Y. When the process is properly
adjusted the characteristics have nominal values Xo and yo, respectively; this results in most of the pro
duction being in the highest priced category q 3 : The process would remain in this state up to the occurence
of disruptive events at t x and t y , which would indicate the beginning of an increase in the values of
X and Y, respectively. In each case, the increase is assumed to follow a known, differentiable monotonic,
invertible function of the time elapsed after the disruptive event. X and Y are thus random variables
with values x and y given by these equations:
x — x<s + h{t — t x ) t>t x
= Xo
y=y +k(tt v ) t>ty
■yo
O^t^t,
An example of the behavior of X and Fis shown in Figure 1.
Table 1
Pricing category
Lower limit
Upper limit
Price received/unit
<7i
L, = °°
f/, = 4.7
«, = $0.70
92
L 2 = 4.7
U z = 4.9
r? 2 = $1.10
<73
L 3 = 4.9
^ = 5.2
R 3 = $3.00
94
L 4 = 5.2
t/ 4 = 5.6
«4= $0.10
95
L 5 = 5.6
t/ 5 = + 00
« 5 = $0.50
785
786
W. J. KENNEDY, JR. AND P. M. GHARE
Vo
4
FIGURE 1. An illustration of the nomenclature used
The disruptive events are assumed to occur at. random times with known probability distributions
each disruptive event independent of the other. After an adjustment both characteristics return to
their nominal values xo and yo, respectively. The objective is to determine an optimal time interval
between adjustments which would maximize the expected net revenue minus the cost of adjustments.
Adjustments are assumed to be performed after equal time intervals.
Problems similar to this have been treated by Pritsker [3], Gibra [1], and Girschick and Rubin
[2]. None of these authors have, however, considered explicitly the case where the quality measure
comes from a probability distribution whose parameters are stochastic, nor have they considered
problems where the price paid is an explicit function of the quality measure. RoelofFs [4] has presented
a method of obtaining qualitydetermined differential pricing places for attributes sampling, but he
has not used his differential pricing plans to determine adjustment intervals.
Symbols and Definitions
X = one characteristic — a random variable
Y= the other characteristic — also a random variable
Xo= the starting, or nominal, value of X
yo= the starting, or nominal, value of Y
OPTIMUM ADJUSTMENT POLICY
787
r a =the time between successive adjustments
7'x=the timeA^ begins to increase, a random variable
T y = the time Y begins to increase, a random variable
fr x (t x ) = the probability density function of T x , evaluated at t x
fr y Uy) = the probability density function of T y , evaluated at t y
h(t — t x ) — the function describing the increase in X as a function of elapsed time since t x
= 0, t^t x
k(t — ty) = the function describing the increase in Y as a function of the elapsed time since t y
= 0,t^ty
fx, y(w, y) = the joint probability massdensity function of X and Y,
A (x) = the value of the elapsed time since t x for which X is equal to x
= h*(x — x ), x < X s * Xo + h(T a )
= 0, x«S xo
B (y ) = the value of the elapsed time since t y for which Y is equal to y
= k 1 (y—y ),yo< y^yo + k(T a )
= 0,y*£yo
The Joint MassDensity Function for X and Y
Since the two characteristics X and Y are random variables, their joint mass density function can
be computed by considering four mutually exclusive and exhaustive cases represented by (xo,yo),
(x , y), (x, y ), and (x, y), respectively.
fx, v(xo, yo) = ( Vr x (t*) £ [lF T (t x )]dt x + f" f T (t y ) £■ [lF Tx (t„)]dt v
Jo X T„ U Jo U T„ x
TOO roo
+ J fr x (t x )[lF Ty (t x )]dt x + fr y (t y )[lFr x (t u )]dt y ;
, , x dA{x) r„4(x) i
fx,y(x,y ) = —— fT x (t I ) — (lFT u [t r + A(x)]dt x Xo<x^x + h(T„);
ax Jo I a
fx, y(x , y) =
1
dy
f frMjl F Tr [t y +B(y)])dty,y <y^ yo + k(T a );
Jo " 1 (,
dB{y)_ rT f ,B(y)
y Jo
4(x) dB(i
dx dy
fx, y(x, y) =,
dA(x) dB(y) f««*> 1
rr a A(x) i
Jo T„
frAt x )frU x + A(x)B(y)]dt x ,
A(x)>B(y)
x < x =£ xo + h{T„)
yo < y^yo + k(T„)
dA{x) dB(y ) f T a «») 1
dx dy
■jfTAty)fT I [ty + B(y)A(x)]dt y
Jo T„ y x
A(x)*B(y)
Xo < x *£ xo + h(T„)
yo<y^yo+k(T„).
788 W. J. KENNEDY, JR. AND P. M. GHARE
In the production process modeled, X and Y are, respectively, the mean and standard deviation
of a normal distribution from which a given measurement is assumed to come. This measurement, a
random variable Z, has the cumulative distribution function given by
(1)
F z (z)=l <f>(w\x ,y )f x ,y(xo,y )dw+\ <j>(w\x,y ) fx, y (x, y )dwdx
J<*> JX J oo
rv +k(T a ) rz
+ 1 <MH* , y)fx, y(x , y)dwdy
Jy J  oo
rx +h(T a ) ry +k(T a ) rz
+ 1 <f>(w\x, y)fx,y)fx, y (x,y) dwdxdy,
Jx Jy J
where <f> (w\x, y) =^=r exp ( ) •
y V2tt L 2 \ y / J
The probability P(qf) that an item falls into pricing category qi is then
P(q*)=Fz(U i )F z (L i ).
The net revenue can be described as the price paid for produced items minus the cost of adjust
ment. The expected net revenue can be expressed as
(2) E [Net revenue] = N £ R t P(q*) C a T R /T a ,
i=l
where
./V= number of items produced in each production run
Ri = per unit revenue in pricing category Pi
C a = cost per adjustment
Tr = duration of each production run
r a =time between successive adjustments
The problem is to find T a so that the net revenue is a maximum.
DETERMINATION OF AN OPTIMUM INTERVAL
An assumption is made on the pricing categories. This assumption is that the pricing category
including the nominal value xo of the process mean has the highest per unit price and prices in other
categories decrease as they move away from xo, i.e.,
xoeqi ^> Rt > Rj,j^i
OPTIMUM ADJUSTMENT POLICY 789
and, fory, k ^ i,
\LjXo\>\L k Xo\ ^> Rj<R k .
It is also assumed that there are at least two pricing categories.
Under these assumptions, and under the previously stated assumptions that A' and Y, as functions
of the time since the last adjustment, are nondecreasing, the price paid for a production lot is a de
creasing function of the time between successive adjustments.
The adjustment cost is a decreasing step function of the time between successive adjustments.
To see this, consider an example where the length of a production run is 31.5 hr. For an interadjustment
time greater than 15.75 and less than 31.5 hr, one adjustment is incurred; between 10.5 and 15.75 hr,
two adjustments are incurred, etc.
The net revenue is thus a discontinuous function of the time T a , with the discontinuities corre
sponding to integer values of TrJTq. As T a increases continuously between these discontinuities, the
net revenue decreases. Hence, the maximum net revenue occurs at a value of T a equal to Tr\ti for some
positive integer n. (If n is equal to 1, the only adjustment is the initial setup at the beginning of the pro
duction run.) Thus, finding the value of T a yielding the maximum net revenue involves a search amongst
the integer values of 7V 7V
NUMERICAL EXAMPLES
As a numerical example let Table 1 represent a pricing scheme and
/V=630
7V=31.5
*o=5.0
yo=0.05
/r x (**) = 0.5exp (0.5 t x )
f Ty (t y ) = 0.1exp(0.1t x )
Fr x (*x) = lexp (0.5 t x )
F T Jty) = l~exp (0.1 t y )
y
h(tt x ) =
k(tt x ) =
0.05 (tt x ) for T a ^t^t x
for s£ t *£ t x
(t^) 2 /100 ioxTa^t^ty
for0=£fs£tj,
790 W. J. KENNEDY, JR. AND P. M. GHARE
I (x 5)10.05 for 5 <* ^5 + 0.05x31.5
A(x) = \
[0 for x *£ 5
f 10 Vy0.05 for0.05< y <0.05 + 0.01 (31. 5) 2
B(y) = \
{ for y< 0.05
C a = 6.00.
For this example, the probability P(q*) is given by
p, *\ fv' 1 r i i»/? 9l lexp[ (0.5 + 0.1)7',]
(3 ) P(«r)  ]y^= exp L" (z*o) 2 /2y 2 ] ^+0OJn ~ *
/•5+0.057 a ft/, 1
+ yo 7 =exp[U^) 2 /2y2]
0.5 exp [0.1(*5)/0.05]
[lexplO^raUSj/COS)}] Q^dzdx
+
fo.os+r^/ioo re, j
J0.05 Jt, V27T
Mp[  (r  5)W] «'exp[WF0^]
[lexp{0.6[r a 10 Vy0.05 ]}1 ; 5 dzdy
L J Vy0.05
f 0.05 + T*/ 100 /"5+0.5V«0.05 ft/, J
J0.05 J 5 J/., V27T
exp[(z*) 2 /2y 2 ] ^^
exp {0.5[10 Vy0.05 (*5)/0.05]}
exp{0.5[7 , a U5)/0.05]
0.1[r„10 Vy0.05]}^ . 5 dzdxdy
0.U5 Vy0.05
+ y^=exp[(z*) 2 /2y 2 ]
J 0.05 J 5+0.5 Vr005 J l { V27T
^ "exp {0. 1 [ (x5)/0.0510 Vy^05 ] }  exp { 0.5k  ^=f*l
 0. 1 [T„  10Vy0.05] } 1 r~ . 5 dzdxdy.
J 0.05 Vy0.05
OPTIMUM ADJUSTMENT POLICY
791
To calculate the optimum T„, T„ is first set to 31.5 (the duration of a production run) and the
probabilities P{q\) calculated from Equation (3) for i equal to 1, . . ., 5. These values of P{q\) are
then substituted into Equation (2), and expected revenue is computed. Next an interadjustment time
corresponding to the next larger integer value of the number of adjustments is used. The process
terminates when the next larger number of adjustment intervals gives a lower revenue than did the last.
To find the maximum net revenue, values of T„ corresponding to integer values of the number of
adjustments were used to calculate the probabilities P{q\). These P(q'j) were then used to calculate
the net revenue for each T„ used. The results are shown in Table 2. In this problem, the maximum
net revenue, $1,795.20 for interadjustment time of 2.86.
TABLE 2. Sample Computations
Number of
adjustments
T„
P(q[)
P(Q' t )
PW 3 )
PW 4 )
P«)
Expected
net
revenue
1
2
3
4
5
6
7
8
9
10
11
12
31.500
15.750
10.500
7.875
6.300
5.250
4.500
3.9388
3.500
3.150
2.864
2.6250
0.1419
0.0443
0.0158
0.0061
0.0024
0.0010
0.0004
0.0002
0.0000
0.0000
0.0000
0.0000
0.0222
0.02533
0.0235
0.0212
0.0194
0.0181
0.0173
0.0169
0.0167
0.0167
0.0168
0.0170
0.2140
0.3954
0.5588
0.6995
0.8077
0.8816
0.9259
0.9506
0.9640
0.9714
0.9755
0.9778
0.2108
0.3593
0.3555
0.2580
0.1643
0.0962
0.0541
0.0304
0.0175
0.0103
0.0063
0.0040
0.4111
0.1756
0.0463
0.0152
0.0061
0.0031
0.0022
0.0019
0.0017
0.0015
0.0014
0.0013
J241.10
706.62
1,061.21
1327.50
1,523.37
1,653.42
1,728.51
1,767.61
1,786.17
1,793.66
1,795.20
1,793.51
REFERENCES
[1 Gibra, Isaac N., "Optimal Control of Processes Subject to Linear Trends," The Journal of Industrial
Engineering, Vol. 18, No. 1 (1967).
[2] Girshick, M. A. and Herman, Rubin, "A Bayes Approach to a Quality Control Model," The Annals
of Mathematical Statistics, Vol. 23 (1952).
[3] Pritsker, A. Alan B., "The Setting of Maintenance Tolerance Limits," The Journal of Industrial
Engineering, Vol. 14 No. 2 (1963).
[4] Roeloffs, Robert, "Acceptance Sampling Plans with Price Differentials," The Journal of Industrial
Engineering, Vol. 18, No. 1 (1967).
SCHEDULING WITH PARALLEL PROCESSORS AND
LINEAR DELAY COSTS
Kenneth R. Baker
North Carolina State University
and
Alan G. Merten
the University of Michigan
ABSTRACT
This paper deals with the sequencing problem of minimizing linear delay costs with parallel
identical processors. The theoretical properties of this mmachine problem are explored, and
the problem of determining an optimum scheduling procedure is examined. Properties of
the optimum schedule are given as well as the corresponding reductions in the number of
schedules that must be evaluated in the search for an optimum. An experimental comparison
of scheduling rules is reported; this indicates that although a class of effective heuristics
can be identified, their relative behavior is difficult to characterize.
1. INTRODUCTION
Two basic facets of scheduling are the allocation of resources and the sequencing of tasks. In
much of the development of scheduling methodology, it has been helpful to simplify the resource struc
ture in order to focus on problems of sequence. Thus in the case of singlemachine finite sequencing
and in the case of jobshop sequencing, it is usually assumed that each requirement for processing in
volves a specified, unique resource (Conway, Maxwell, Miller [5]). A first step in treating resource
flexibility then is to deal with one resource type and parallel capability. This study deals with the prob
lem of minimizing mean weighted flowtime with parallel machines and with independent nonpreemp
table tasks.
The model to be considered involves m machines and n jobs. The machines are identical and each
is capable of processing at most one job at a time. The n jobs are independent (that is, no precedence
relations exist among them), simultaneously available at time zero, and can each be processed by at
most one machine at a time. In addition, job / has associated with it a processing time (denoted Pj),
known in advance, and a weighting factor (wj), reflecting its value or importance.
For job./, the flowtime (Fj) denotes the time spent in the system until completion. The performance
measure of interest is meanweighted flowtime:
i »fs
Another way of looking at the same problem is to suppose that each job has an associated delay
cost per unit time spent in the system (McNaughton [15]). If w } denotes this unit delay cost, then under
793
794 K. R. BAKER AND A. G. MERTEN
a given schedule the total delay cost accumulated for job j is WjFj. The problem of minimizing total
delay cost for the set of n jobs is identical to the problem of minimizing F w .
For m = 1, the wellknown result is that F w is minimized by processing the jobs in nondecreasing
order of the ratio Pj/tVj. This ordering will be referred to as weighted shortest processing time (WSPT)
sequencing. The sequence which maximizes F w is antithetical sequence, weighted longest processing
time (WLPT). For m > 1, comparable results do not exist, largely due to the presence of a resource
allocation problem superimposed upon the sequencing problem.
Models involving F w as a performance measure have been employed in a diverse set of applications,
as represented by several of the references (Riesel [22], Merten [17], Bowdon [3], Coffman and Muntz [4],
Baker [2], and Grieshop [10]). Some of these sources treat models which explicitly contain parallel
processors. In those cases where the discussion is limited to single processor models, it is not difficult
to recognize that the parallel processor case is an important and realistic extension of the specialized
model. Therefore, the scheduling problem treated in this paper has a very broad spectrum of potential
application areas.
2. THEORETICAL RESULTS FOR THE mMACHINE PROBLEM
The search for the sequence that minimizes mean flowtime and mean waiting time theoretically
must consider all possible schedules of the n jobs on the m machines. This search would be based on
the relative weights and processing times of the jobs just as it is in the onemachine case. It is possible,
however, to reduce the number of schedules that must be investigated because of certain dominance
properties, properties of the mean as a performance function, and symmetries. A schedule of the
n jobs on the m machines can be viewed as taking place in the following phases:
(1) partition the n jobs into m sets (machines)
(2) order the jobs on each of the m machines
Because of the linear properties of the mean, the mean flowtime for the n jobs on the m machines
is the sum of the mean times for each of the m machines. Therefore, to minimize the mean flowtime
over all machines, the mean flowtime must be minimized on each machine (Eastman, Even, and
Isaacs [7]). Given that a subset of the jobs is assigned to a particular machine, the optimum sequence
for this machine corresponds to the WSPT ordering. Therefore, the number of schedules that must be
investigated is no greater than the number of ways of assigning n jobs to m machines since, from the
onemachine results, WSPT is known to be optimal on a single machine.
The next reduction in the size of the set of schedules to be investigated comes as a result of the
observation that an optimum schedule cannot contain an empty machine. For if a schedule were to
include an empty machine, a schedule which has a lower mean flowtime can be obtained by taking a
job from a machine where there is more than one job and moving it to the previously empty machine
(the waitingtime for the job that is moved is reduced from a positive number to zero). The process of
eliminating empty machines can be continued until there are no empty ones left.
Finally, since the machines are identical prior to the assignment of the jobs, certain schedules
can be ignored since they are indistinguishable from other schedules.
In order to describe sets of schedules, let
Z = set of all possible ways of scheduling n jobs on m machines;
Z p — set of all possible ways of scheduling n jobs on m machines if the sequence within a machine
is ignored (use the WSPT sequence within each machine);
PARALLEL PROCESSORS SCHEDULING
795
Z e = set of all possible ways of scheduling n jobs on m machines using WSPT for each machine
and excluding the cases where there is one or more unused machines;
Z m = set of all possible ways of scheduling n jobs on m machines using WSPT, excluding the
unused machine cases and ignoring the indistinguishable schedules.
Therefore, Z D Z p D Z e D Z m . The problem is now to find N(A), the number of elements in each of
these sets A.
Table 1 shows the expressions for the sizes of the sets defined above and gives examples for some
small values of n and m. The derivation of these expressions is given in Merten [17]. Some of these
results were derived from previous work in combinatorial analysis (Feller [8]. Knuth [14J, and Abramovitz
and Stegun [1]). Even for these small numbers of jobs and machines, it is clearly important to isolate
the subset of schedules that contains the sequence that minimizes mean waitingtime.
The following additional results have been shown for the general mmachine problem:
1. It is sufficient to consider schedules in which there is no preemption of jobs (McNaughton [15).
2. A lower bound B{m), on the optimum solution can be obtained as follows (Eastman, Even,
Issacs [7J): Let B{\) denote optimum value of F w for the given job set when m= 1 (given by WSPT).
Let B{n) denote the optimum value of F w when m — n (given by assigning each job to a different
machine, so that Fj = Pj). Then
B{m) = max fi(/i),fl(l) + ^—  B(n)\.
[ m 2m J
3. The problem can be formulated as a dynamic programming problem (Held and Karp [11 J).
This formulation can include the case where the job execution time may differ depending on which
machine is used. While the dynamic programming formulation does lead to some reduction in the
computation required to find the optimum solution, the procedure is still inadequate for solving large
problems.
Table 1. Number of Ways of Sequencing njobs on m Machines as a Function of the Schedule Set
N(Z)
N(Z P )
N(Z t )
N(Z m )
(n, m)
"(TV)
m"
{><
n
(3,2)
24
8
6
3
(4,2)
120
16
14
7
(3,3)
60
27
6
1
(4,3)
360
81
36
6
When all weights are equal, the optimal schedule for the mmachine problem can be constructed
by arranging the jobs in nondecreasing order of processing time and then assigning the jobs in this order
to a machine as soon as one is made available. In practice, this would correspond to the creation of a
796
K. R. BAKER AND A. G. MERTEN
single jobqueue in which shortestfirst priority prevails. Whenever a machine became available, it
would be assigned the highest priority job among those remaining in the queue.
Several alternate constructions will yield optimum solutions as well, and it is useful to consider
a different viewpoint. As discussed by Conway, Maxwell, and Miller [5, pp. 7778] an optimal schedule
for the equal weighting problem can be found as follows:
1. Find the jobs with the m longest processing times and assign them in any order to m different
machines.
2. Remove the assigned jobs from consideration and repeat step 1 until all jobs are assigned.
3. At each machine, process the jobs in shortestfirst sequence.
Similarly, it has been shown, when all the job processingtimes are equal, the optimal allocation
has the property that the m jobs with the largest weights are in the first positions on the m machines,
the jobs with the m largest weights of those remaining are in the second positions on the m machines,
and so on until all the jobs have been assigned a position. (Merten [17]).
3. HEURISTIC SCHEDULING PROCEDURES
The foregoing discussion of the equalweighting problem serves to identify two basic approaches
to the more general problem: a oneatatime job assignment strategy and an matatime job assignment
strategy.
Under the oneatatime strategy, which is called heuristic Hi, a priority rule is selected in order
to form a ranked fist of the jobs. The machine with the smallest amount of scheduled processing is
then assigned the first job on the list. This step is repeated until all jobs are assigned to machines and
then the jobs assigned to each machine are ordered by WSPT. To illustrate how heuristic Hi works,
consider the 10job set shown in Table 2 and suppose that five machines are available. Also, suppose
that the priority rule WLPT is selected to form the initial ranked list (so that the jobs are considered in
the order 10, 9, 8, . . .,2, 1.) At the start, no processing has been assigned to any machine, so the first
five jobs on the list are assigned to five different machines (see Table 3). At this stage, the vector of
total processing commitments assigned to each machine is (22, 32, 41, 50, 19). Since the minimum occurs
for machine 5, the next job (job 5) is assigned to machine 5. The updated vector of machine commit
ments is (22, 32, 41, 50, 45). Now the minimum occurs for machine 1, and so the next job (job 4) is
assigned to machine 1. The details of the procedure are presented in Table 3, and the final sequence
generated by this combination of H\ and WLPT is shown in Figure 1. If WLPT were replaced by some
other priority rules for ranking the jobs initially, H\ might lead to a different schedule, with a different
value of F w .
Table 2. A 10Job Data Set, in which the Jobs Are Numbered in WSPT Order
Job
1
2
3
4
5
6
7
8
9
10
Pi
5
21
16
6
26
19
50
41
32
22
Wj
4
5
3
1
4
2
5
4
3
2
PilWj
1.25
4.2
5.3
6
6.5
9.5
10
10.2
10.7
11
Under the basic matatime strategy, which is called heuristic H m , a priority rule is again selected
to form a ranked list of the jobs. The first m jobs on the list are assigned to m different machines. The
PARALLEL PROCESSORS SCHEDULING
797
1
3
1
4
1
10
1
16
22
44
2
2
1
9
1
21
53
3
' 1
8
~l
5
41
4
50
5
5
1
6
1
26
H,(WLPT); F w = 32.42
Figure 1.
45
Table 3
1. Initial job list {10,9,8,7,6,5,4,3,2,1}
2. Assignment phase.
Processing commitments
Job Machine assinged
(0,0,0,0,0)
10
1
(22,0,0,0,0)
9
2
(22,32,0,0,0)
8
3
(22,32,41,0,0)
7
4
(22,32,41,50,0)
6
5
(22,32,41,50,19)
5
5
(22,32,41,50,45)
4
1
(28,32,41,50,45)
3
1
(44,32,41,50,45)
2
2
(44,53,41,50,45)
1
3
3. Reorder at each machine by WSPT
Machine Sequence
1 3410
2 29
3 18
4 7
5 56
next m jobs on the list are assigned to m unique machines and so on, until all jobs are assigned. Then
WSPT sequencing is applied to each machine. The mway assignment required at each stage can be
specified in more detail. Consider the situation in which the assignment step has been repeated s times,
so that sm jobs are assigned. Taking these assignments to be fixed, consider the subproblem in which
it is desired to allocate the next m jobs at stage 5 + 1 so that mean weighted flowtime is minimized for
all assigned jobs. It is not difficult to show that the optimum allocation in this subproblem is the assign
ment of the job with the largest weighting factor to the machine with the next smallest processing com
mitment, and so on. This assignment mechanism is incorporated in //,„. To illustrate this heuristic,
consider the example introduced above, and again let WLPT be used to rank the jobs initially. At the
first stage, jobs 610 are assigned to different machines (see Table 4); this yields a machine commit
798
K. R. BAKER AND A. G. MERTEN
Table 4.
1. Initial job list
{10,9,8,7,6,5,4,3,2,1}
2. Assignment phase
Stage
Processing commitments
7o6s(wj)
Machine assigned
1
(0,0,0,0,0)
10(2)
1
9 (3)
2
8(4)
3
7(5)
4
6(2)
5
2
(22,32,41,50,19)
5(4)
1
4(1)
4
3(3)
3
2(5)
5
1(4)
2
3. Reorder at each machine by WSPT
Machine
Sequence
1
510
2
19
3
38
4
47
5
26
merits vector of (22, 32, 41, 50, 19). At the second stage the machines are ordered smallestfirst by this
commitment (51234) and the jobs are ordered largestfirst by weighting factor, ties being broken
arbitrarily (25134). This leads to the assignment of job 2 to machine 5, job 5 to machine 1, job 1 to
machine 2, job 3 to machine 3, and job 4 to machine 4. The details of this heuristic are given in Table 4
and Figure 2. Once again, the use of a priority rule different from WLPT might lead to a different sched
ule. In any case, when n is an even multiple of m (as in the example problem), heuristic H m will always
assign the same number of jobs to every machine.
One variation of this form of H m is to relax the restriction that the m jobs must be assigned to differ
ent machines at each stage. In this case, a possible heuristic is to treat the m jobs in decreasing order
of their weighting factors and to assign them one at a time to the machine with the smallest processing
10
26
48
n
37
16
57
A_L
56
21 40
HJWLPT), F w = 32.67
Figure 2.
PARALLEL PROCESSORS SCHEDULING
799
commitment. Under this heuristic, called H x , it is possible that several of the m jobs considered at a
given stage will be assigned to the same machine. We were surprised, however, to find in our experi
mentation that H x was consistently less effective than the other two heuristics.
The priority rule at the heart of each heuristic can be selected from a variety of orderings which are
potentially effective in sequencing. At least five basic priority rules are of interest: shortest processing
time (SPT), longest processing time (LPT), their weighted versions (WSPT and WLPT), and largest
weighting factor (W). Used in conjunction with the three heuristics H u H m , and H x , they yield 15 dis
tinct scheduling procedures with various combinations being denoted as Hi (WLPT), H m (W), etc. The
remainder of the discussion deals with the solutions generated by these procedures.
For the 10job example of Table 2, an examination of the 15 scheduling procedures reveals that the
best schedule is not one of those in Figures 13, but is one produced by H,„ (WSPT), with F w = 32.30.
(In order to determine an optimum schedule, it would be necessary to examine N(Z m ) =42,525 sched
ules in this problem.) The second best schedule is produced by Hi (WLPT) and the third best by Hi
(LPT). It is interesting to observe, however, that for the same job set with a different number of machines,
the relative ordering of the scheduling procedures is somewhat different. Table 5 displays results for
2 =£ m ^ 6. Two important properties are evident: first, several different procedures produce best sched
ules at least once in the five different problems; and second, the set of three best procedures is differ
ent for every value of m. There is no clearcut choice for the best scheduling rule, nor is there yet even
a convincing choice between heuristics Hi and H m .
This instability of relative performance among scheduling procedures might well be particularly
characteristic of small problems. When n is small, a change in the scheduling of one or two jobs can
represent a significant change in the overall performance measure, whereas this is much less likely to
be the case when n is large. As illustrated by this example problem, when the job set is small there may
be considerable nonuniformity in the effectiveness of a particular rule. It is doubtful that a truly opti
mum procedure will exist among the heuristic procedures examined here.
To shed some light on the question of the effect of problem size, a more detailed investigation of
larger problems was carried out.
Table 5. Rankings of the Three Best Rules for Different Values ofm.
m
2
3
4
5
6
Best
H, (WSPT) a
Hi (LPT)
Hi (WSPT)
W m (WSPT)
Wm(W) a
Second
H m (WSPT) a
// m (WSPT)
Hi (WLPT)
Hi (WLPT)
H,wr
Third
tf,(WSPT)
Hi (SPT)
W m (W)
Hi (LPT)
Hi (WSPT) a
' Ties.
4. EXPERIMENTS WITH LARGE JOB SETS
Six large job sets, of size n= 100 jobs, were constructed for further experimentation. The job sets
were generated as follows: (a) the processing times in each set consisted of random samples drawn
from a distribution with a mean of 50, (b) the weights in each set were independent of the processing
800
K. R. BAKER AND A. G. MERTEN
times and were samples drawn from a distribution with a mean of five. The job sets were distinguished
by the forms of the distribution, as shown in Table 6. For each job set, the same jobs were scheduled
for m parallel machines, where m was again varied from 2 to 6. With six data sets and five versions of
parallelism, 30 different problems were posed, and each of the 15 scheduling procedures was tested on
each of the 30 problems.
Detailed results for data set four are displayed in Table 7. For each combination of scheduling pro
cedure and number of machines, the schedule value is given as the difference between F w and the lower
bound, along with its ranking among the 15 procedures. This particular data set is perhaps a typical
among the six tested, but it does serve to highlight many of the characteristics of the problem. Specifi
cally, the results show that there is only a limited amount of dependence in the rankings as m is varied.
Although the five different values of m do not generate completely independent sets of observations,
they do convey much more information than the results for any single value of m alone.
Three different rules emerged as best in this job set: H m (WSPT), H x (WLPT), and //, (WSPT).
Thus two different heuristics Hi and H,„ emerged as best. This configuration dramatically illustrates
Table 6
Data Set
1
2
3
4
5
6
/< distribution
U(0, 100)
/V(50, 10)
f/(0, 100)
/V(50, 10)
t/(0, 100)
*(50)
(/'■distribution
U(0, 10)
U(0, 10)
/V(5, 1)
N(5, 1)
X(5)
*(5)
Notation: U(a, b) Uniform on the interval a to 6.
N(a, b) Normal with mean a and standard deviation b.
X(a) Exponential with mean a.
Table 7
m
2
3
4
5
6
//, (SPT)
0.65
8
1.23
9
1.91
9
2.39
9
1.55 5
(WSPT)
0.10
2
0.18
2
0.18
1
0.31
2
0.29 1
(LPT)
2.04
11
1.32
10
1.55
8
1.90
7
2.43 11
(WLPT)
0.12
4
0.16
1
0.19
2
0.35
4
0.33 2
iW)
0.51
7
3.02
13
2.18
11
2.17
8
1.94 8
H m (SPT)
20.89
14
15.91
14
13.91
14
12.07
15
11.98 14
(WSPT)
0.09
1
0.22
3
0.22
3
0.27
1
0.38 3
(LPT)
21.53
15
16.40
15
14.62
15
11.18
14
12.80 15
(WLPT)
0.11
3
0.23
4
0.27
4
0.34
3
0.17 4
(r)
2.12
12
0.71
7
1.44
6
2.45
10
1.94 8
H x (SPT)
5.88
13
2.27
11
2.92
12
3.92
13
4.29 13
(WSPT)
0.36
6
0.31
5
1.48
7
1.02
6
2.21 10
(LPT)
0.98
9
1.22
8
4.33
13
2.64
11
3.14 12
(WLPT)
0.27
5
0.32
6
0.57
5
0.88
5
1.85 7
(W)
1.22
10
2.74
12
1.96
10
2.79
12
1.68 6
The numbers shown for each combination are (1) the difference between F w and the lower bound and (2) the rank of the
F w value among the rules tested.
PARALLEL PROCESSORS SCHEDULING
801
that no single heuristic will always be associated with the best schedule and that no single priority rule
will always be associated with the best schedule.
Secondly, the rankings indicated quite clearly that the weighted priorities are more effective
than their unweighted counterparts. Only under H x did a weighted priority lead to a rank below sixth.
Thirdly, H x did not produce a schedule ranked better than fifth and was clearly worse than the
other heuristics. Presumably, H x suffers from the fact that it does not necessarily distribute the jobs in
equal numbers among machines.
The heuristic H„„ by contrast, is restricted to distributing the jobs in equal numbers among
machines. While this characteristic is sometimes favorable, it is distinctly unfavorable in the case of
the unweighted priorities, which ranked 14 and 15 in all six problems.
Finally, there appears to be no overall clear choice between the priorities WSPT and WLPT. For
H m , WSPT seems to be uniformly more effective, but for H\ no similar conclusion can be drawn. In
some respects, this may be the most surprising property illustrated by these results, for although WLPT
maximizes F w for m—1, it can be incorporated into the parallel processor case in a desirable way.
The important results in the six data sets are summarized in Table 8 by the use of rankings where
the specific rules which produced the three best schedules are shown in ranked order for all 30 problems.
Of the 30 outcomes, the distribution of best schedules was as follows:
Best
tf,(WSPT) 21
/7 m (WSPT 6
#i(WLPT) 3
Second
Third
Total
9
30
5
17
28
16
8
27
Table 8. Comparison of Rules
m=
2
3
4
5
6
DSl
Ht (WSPT)
//, (WLPT)
H m (WSPT)
Hi (WSPT)
Hi (WLPT)
Hm (WSPT)
Hi (WSPT)
Hi (WLPT)
H m (WSPT)
Hi (WSPT)
Hi (WLPT)
Hm (WSPT)
Hi (WSPT)
Hi (WLPT)
tf m (WSPT)
DS2
Ht (WSPT)
Hi (WLPT)
H m (WSPT)
Hi (WSPT)
Hm (WSPT)
Hi (WLPT)
Hi (WSPT)
Hi (WLPT)
H m (WSPT)
Hi (WSPT)
Hi (WLPT)
Hm (WSPT)
Hi (WSPT)
// m (WSPT)
H (WLPT)
DS3
Hi (WSPT)
H (WLPT)
Hm (WSPT)
Hi (WSPT)
Hi (WLPT)
Hm (WSPT)
Hi (WSPT)
Hm (WSPT)
Hi (WLPT)
Hi (WSPT)
Hi (WLPT)
Hm (WSPT)
//, (WLPT)
//, (WSPT)
Hm (WLPT)
DSl
H m (WSPT)
Hi (WSPT)
H m (WLPT)
Hi (WLPT)
H (WSPT)
H m (WSPT)
Hi (WSPT)
Hi (WLPT)
Hm (WSPT)
Hm (WSPT)
Hi (WSPT)
H m (WLPT)
W, (WSPT)
Hi (WLPT)
tf m (WSPT)
DSS
Hm (WSPT)
Hi (WSPT)
H m (WLPT)
Hi (WLPT)
Hi (WSPT)
H m (WSPT)
Hi (WSPT)
Hm (WSPT)
Hi (WLPT)
Hi (WSPT)
H m (WSPT)
Hi (WLPT)
A/, (WSPT)
H m (WSPT)
tf m (WLPT)
DS6
Hi (WSPT)
Hi (WLPT)
H m (WSPT)
H n (WSPT)
Hi (WSPT)
Hi (WLPT)
Hi (WSPT)
Hi (WLPT)
Hm (WSPT)
H m (WSPT)
Hi (WSPT)
Hi (WLPT)
H m (WSPT)
W, (WSPT)
W, (WLPT)
802 K. R. BAKER AND A. G. MERTEN
Hi (WSPT) most frequently produced the best schedule and always produced one of the two best sched
ules. H m (WSPT) and Hi (WLPT) were less likely to produce the best schedule, but nearly as likely to
produce one of the three best. While the size of the problem precludes a comparison of the best heuristic
solution with the true optimum, we observed that Hi (WSPT) was within 1 percent of the lower bound
95 percent of the time. Therefore, it appears that only very slight improvements could possibly be made
over the solution obtained with this heuristic procedure.
5. CONCLUSIONS
If an optimum rule for this problem exists (that is, a scheduling mechanism more efficient than
enumeration) it is likely to be quite complicated. Furthermore, extensions to flowtime problems with
multiple resource types or with nonstatic job arrivals would also appear to be complex.
The primary element in attempting to minimize F w with parallel processors is the use of the condi
tion that WSPT should prevail for each processor. This condition is so important to nearoptimal sched
uling that only marginal improvements can be expected from sophisticated assignments of jobs to
machines. Moreover, this investigation suggests that the relative behavior of heuristic procedures for
this assignment process may be extremely difficult to characterize in general. The prospect is that spe
cial problem attributes (distribution of processing times, number of machines, etc.) will affect the per
formance of different procedures and perhaps render the concept of an "optimum rule" meaningless.
The results for the large job sets indicate that H x is the least effective of the three heuristics tested
and that neither Hi nor H m is consistently best. Indeed, for a given set of jobs, it is possible that the rela
tive performance of Hi and H m is reversed as m is varied. Nevertheless, Hi did appear to be perceptibly
more likely than H m to produce the best schedule. The outcome is a pleasant surprise in that Hi is the
simplest of the three heuristics to implement and H x is the most difficult.
In much the same way, no priority ordering was consistently best, although it was clear that
weighted priorites were more reliable than unweighted priorities. The effectiveness of WLPT might
be attributable to the fact that longestfirst sequencing tends to distribute processing fairly equally
among machines, as discussed by Kedia [13] and illustrated in Figure 1. Nevertheless, WSPT appeared
to be the best priority ordering.
The fact that both WSPT and WLPT were effective might suggest that weighting factors are more
important job traits than processing times. Yet the largestweight priority was unable to produce one
of the three best schedules in any of the 30 problems.
With regard to the effect of processing time distributions and weighting factor distributions, the
results are inconclusive. If anything, Hi (WSPT) was most effective when the weights were uniformly
distributed and was least effective for data sets 4 and 6, but these represented the job sets with the
least and most variability. More testing would be required to determine whether there is a significant
distribution effect. From the limited scope of these results, however, one might infer that the conclusions
hold for a wide variety of distributions.
Granting the lack of consistency which is inherent in the problem, the data in Table 8 certainly
recommend Hi (WSPT) as the most effective scheduling procedure. In addition to the high frequency
with which it produced good schedules, Hi (WSPT) has other advantages. First, it is a logical rule to
use, since it is a generalization of the optimum rule for the multiprocessor problem with equal weights.
Secondly, it is a onepass procedure, and does not require a reordering of the jobs once they have been
assigned to machines. It is slightly simpler than H m (WSPT), which includes an additional assignment
mechanism at each stage, and is probably the simplest procedure of those studied. Finally, Hi (WSPT)
PARALLEL PROCESSORS SCHEDULING 803
structurally is a dispatching procedure: the final job assignments are made at chronologically ordered
points in time (i.e., in the order they would be implemented.) This type of structure is likely to be more
adaptable as part of a larger, more complex problem than twopass procedures or iterative schemes. In
particular, problems with multiple resource types or with dynamic job arrivals are important extensions
of the problem considered here, and they can accomodate the H\ (WSPT) heuristic without major
obstacles.
BIBLIOGRAPHY
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[5] Conway, R. W., W. L. Maxwell and L. W. Miller, Theory of Scheduling (AddisonWesley, Reading,
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[6] Denby, D. C, "Minimum Downtime as a Function of Reliability and Priority Assignments in
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[8] Feller, W. "An Introduction to Probability Theory and Its Applications" (John Wiley & Sons,
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University of Wisconsin Computing Center (June 1970).
804 K. R. BAKER AND A. G. MERTEN
[18] Mitten, L. G., "An Analytic Solution to the LeastCost Testing Sequence Problem," Journal of
Industrial Engineering, Vol. 11, No. 1 (Jan Feb. 1960), p. 17.
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(1956).
NEWS AND MEMORANDA
Logistics Research Conference
The Office of Naval Research and the George Washington University, with the cooperation of
the Air Force Office of Scientific Research and the Army Research Office, announce a Logistics
Research Conference to be held at the George Washington University, Washington, D.C., on 8 10
May 1974. The main objectives of the Conference are to survey major developments and difficulties
in government, industrial and military logistics research and applications since World War II, and to
assess outstanding current problems and promising new research techniques.
Areas of research activity have been categorized, as follows: 1) applications of mathematical
programming; 2) applied case studies; 3) design of systems; 4) inventory systems; 5) data collection,
representation, and analysis; 6) measurement of performance; 7) probabilistic methods; 8) production
and procurement; 9) reliability, maintainability, and availability; 10) simulation; 11) statistical methods;
and 12) transportation and scheduling.
Contributed papers are welcome. Abstracts and inquiries may be addressed to Ms. Henrietta
Jones, Department of Operations Research, The George Washington University, Washington, DC
20006. (Phone: 202/6767504.) Further information may also be obtained from Professors Anthony V.
Fiacco (202/6767511), W. H. Marlow (202/6767503), or Henry Solomon (202/6767521) at the Uni
versity, or from Mr. Marvin Denicoff (202/6924304) at the Office of Naval Research.
805
INDEX TO VOLUME 20
AGRAWAL, A. C. and L. L. George, "Estimation of a Hidden Service Distribution of an Af/G/°° System,"
Vol. 20, No. 3, Sep. 1973, pp. 549555.
AHSANULLAH, M. and M. A. Rahim, "Simplified Estimates of the Parameters of the Double Ex
ponential Distribution Based on Optimum Order Statistics from a MiddleCensored Sample,"
Vol. 20, No. 4, Dec. 1973, pp. 745751.
ALAM, K., "Peak Rate of Occurrence of a Poisson Process," Vol. 20, No. 2, Jun. 1973, pp. 269275.
BAKER, K. R., and A. G. Merten, "Scheduling with Parallel Processors and Linear Delay Costs,"
Vol. 20, No. 4, Dec. 1973, pp. 793804.
BAZARAA, M. S., "Geometry and the Resolution of Duality Gaps," Vol. 20, No. 2, Jun. 1973, pp.
357366.
BAZARAA, M. S., D. C. Montgomery and A. K. Keswani, "Inventory Models with a Mixture of Back
orders and Lost Sales," Vol. 20, No. 2, Jun. 1973, pp. 255263.
BENNETT, G. K. and H. F. Martz, "An Empirical Bayes Estimator for the Scale Parameter of the
TwoParameter Weibull Distribution," Vol. 20, No. 3, Sep. 1973, pp. 387393.
BOL, G. and O. Moeschlin, "Applications of Mills' Differential," Vol. 20, No. 1, Mar. 1973, pp. 101108.
BRATLEY, P., M. Florian and P. Robillard, "On Sequencing with Earliest Starts and Due Dates with
Application to Computing Bounds for the (n/m/G/F max ) Problem," Vol. 20, No. 1, Mar. 1973,
pp. 5767.
BROWN, G. F. and W. F. Rogers, "A Bayesian Approach to Demand Estimation and Inventory Pro
visioning," Vol. 20, No. 4, Dec. 1973, pp. 607624.
BROWN, G. G. and H. C. Rutemiller, "A Cost Analysis of Sampling Inspection under Military Standard
105D," Vol. 20, No. 1, Mar. 1973, pp. 181199.
BURDET, C, "Polaroids: A New Tool in NonConvex and in Integer Programming," Vol. 20, No. 1,
Mar. 1973, pp. 1324.
BURTON, R. W. and S. C. Jaquette, "The Initial Provisioning Decision for Insurance Type Items,"
Vol. 20, No. 1, Mar. 1973, pp. 123146.
BUTTERWORTH, R. W. and T. Nikolaisen, "Bounds on the Availability Function." Vol. 20, No. 2
Jun. 1973, pp. 289296.
CALLAHAN, J. R., "A Queue with Waiting Time Dependent Service Times," Vol. 20, No. 2, Jun.
1973, pp. 321324.
CASSIDY, R. G., C. A. Field, and M. J. L. Kirby, "Partial Information in Two Person Games with
Random Payoffs," Vol. 20, No. 1, Mar. 1973, pp. 4156.
CHARNES, A. and W. W. Cooper, "An Explicit General Solution in Linear Fractional Programming,"
Vol. 20, No. 3, Sep. 1973, pp. 449467.
COOPER, W. W. and A. Charnes, "An Explicit General Solution in Linear Fractional Programming,"
Vol. 20, No. 3, Sep. 1973, pp. 449467.
CRAVEN, B. D. and B. Mond, "A Note on Mathematical Programming with Fractional Objective
Functions," Vol. 20, No. 3, Sep. 1973, pp. 577581.
CROW, R. T., "An Approach to the Allocation of Common Costs of MultiMission Systems," Vol. 20,
No. 3, Sep. 1973, pp. 431447.
807
808 INDEX OF VOLUME 20
CUNNINGHAM, A. A. and S. K. Dutta, "Scheduling Jobs, with Exponentially Distributed Processing
Times, on Two Machines of a Flow Shop," Vol. 20, No. 1, Mar. 1973, pp. 6981.
DENT, W., R. Jagannathan and M. R. Rao, "Parametric Linear Programming: Some Special Cases,"
Vol. 20, No. 4, Dec. 1973, pp. 725728.
DUDEWICZ, E. J. and C. Fan, "Further Light on Nonparametric Selection Efficiency," Vol. 20, No. 4,
Dec. 1973, pp. 737744.
DUTTA, S. K. and A. A. Cunningham, "Scheduling Jobs, with Exponentially Distributed Processing
Times, on Two Machines of a Flow Shop," Vol. 20, No. 1, Mar. 1973, pp. 6981.
EICHHORN, B. H., "Sequential Search of an Optimal Dosage: NonBayesian Methods," Vol. 20,
No. 4, Dec. 1973, pp. 729736.
FAN, C. and E. J. Dudewicz, "Further Light on Nonparametric Selection Efficiency," Vol. 20, No. 4,
Dec. 1973, pp. 737744.
FENSKE, W. J. and S. Zacks, "Sequential Determination of inspection Epochs for Reliability Systems
with General Lifetime Distributions, "Vol. 20, No. 3, pp. 377386.
FIELD, C. A., R. G. Cassidy and M. J. L. Kirby, "Partial Information in Two Person Games with
Random Payoffs," Vol. 20, No. 1, Mar. 1973, pp. 4156.
FLORIAN, M., P. Bratley and P. Robillard, "On Sequencing with Earliest Starts and Due Dates with
Application to Computing Bounds for the (ra/m/G/F max ) Problem," Vol. 20, No. 1, Mar. 1973,
pp. 5767.
GEORGE, L. L. and A. C. Agrawal, "Estimation of a Hidden Service Distribution of anM/G/°° System,"
Vol. 20, No. 3, Sep. 1973, pp. 549555.
GHARE, P. M. and W. J. Kennedy, Jr., "Optimum Adjustment Policy for a Product with Two Quality
Characteristics," Vol. 20, No. 4, Dec. 1973, pp. 785791.
GRUNSPAN, M. and M. E. Thomas. "Hyperbolic Integer Programming," Vol. 20, No. 2, Jun. 1973,
pp. 341356.
GUSTAFSON, S. A. and K. O. Kortanek, "Numerical Treatment of a Class of SemiInfinite Program
ming Problems," Vol. 20, No. 3, Sep. 1973, pp. 477504.
HARTMAN, J. K., "Some Experiments in Global Optimization," Vol. 20, No. 3, Sep. 1973, pp. 569
576.
HEIDER, C. H., "An NStep, 2Variable Search Algorithm for the Component Placement Problem,"
Vol. 20, No. 4, Dec. 1973, pp. 699724.
HEIMANN, D. and M. F. Neuts. "The Single Server Queue in Discrete TimeNumerical Analysis IV,"
Vol. 20, No. 4, Dec. 1973, pp. 753766.
HENIN, C. G., "Optimal Allocation of Unreliable Components for Maximizing Expected Profit Over
Time," Vol. 20, No. 3, Sep. 1973, pp. 395403.
HOCHBERG, M., "Generalized Multicomponent Systems under Cannibalization," Vol. 20, No. 4,
Dec. 1973, pp. 585605.
HOWES, D. R. and R. M. Thrall, "A Theory of Ideal Linear Weights for Heterogeneous Combat
Forces," Vol. 20, No. 4, Dec. 1973, pp. 645659.
JAGANNATHAN, R., W. Dent and M. R. Rao, "Parametric Linear Programming: Some Special
Cases," Vol. 20, No. 4, Dec. 1973, pp. 725728.
JAQUETTE, S. C. and R. W. Burton, "The Initial Provisioning Decision for Insurance Type Items,"
Vol. 20, No. 1, Mar. 1973, pp. 123146.
INDEX OF VOLUME 20 809
KALYMON, B. A., "Structured Markovian Decision Problems," Vol. 20, No. 1, Mar. 1973, pp. 111.
KAPLAN, A. J., "A Stock Redistribution Model," Vol. 20, No. 2, Jun. 1973, pp. 231239.
KAPLAN, S., "Readiness and the Optimal Redeployment of Resources," Vol. 20, No. 4, Dec. 1973,
pp. 625638.
KAPUR, K. C, "On MaxMin Problems," Vol. 20, No. 4, Dec. 1973, pp. 639644.
KENNEDY, W. J., Jr. and P. M. Ghare, "Optimum Adjustment Policy for a Product with Two Quality
Characteristics," Vol. 20, No. 4, Dec. 1973, pp. 785791.
KESWANI, A. K., M. S. Bazaraa and D. C. Montgomery, "Inventory Models with a Mixture of Back
orders and Lost Sales," Vol. 20, No. 2, Jun. 1973, pp. 255263.
KHUMAWALA, B. M., "An Efficient Heuristic Procedure for the Uncapacitated Warehouse Location
Problem," Vol. 20, No. 1, Mar. 1973, pp. 109121.
KIRBY, M. J. L., C. A. Field and R. G. Cassidy, "Partial Information in Two Person Games with
Random Payoffs," Vol. 20, No. 1, Mar. 1973, pp. 4156.
KLIMKO, E. and M. F. Neuts, "The Single Server Queue in Discrete TimeNumerical Analysis II,"
Vol. 20, No. 2, Jun. 1973, pp. 305319.
KLIMKO, E. and M. F. Neuts, "The Single Server Queue in Discrete TimeNumerical Analysis III,"
Vol. 20, No. 3, Sep. 1973, pp. 557567.
KORTANEK, K. O. and S. A. Gustafson, "Numerical Treatment of a Class of SemiInfinite Program
ming Problems," Vol. 20, No. 3, Sep. 1973, pp. 477504.
LANGFORD, E., "A Continuous Submarine Versus Submarine Game," Vol. 20, No. 3, Sep. 1973,
pp. 405417.
MARTZ, H. F. and G. K. Bennett, "An Empirical Bayes Estimator for the Scale Parameter of the
TwoParameter Weibull Distribution" Vol. 20, No. 3, Sep. 1973, pp.
MASTRAN, D. V. and C. J. Thomas, "Decision Rules for Attacking Targets of Opportunity," Vol. 20,
No. 4, Dec. 1973, pp. 661HS72.
MERTEN, A. G. and K. R. Baker, "Scheduling with Parallel Processors and Linear Delay Costs,"
Vol. 20, No. 4, Dec. 1973, pp. 793804.
MOESCHLIN, O. and G. Bol, "Applications of Mills' Differential," Vol. 20, No. 1, Mar. 1973, pp.
101108.
MOND, B. and B. D. Craven, "A Note on Mathematical Programming with Fractional Objective
Functions," Vol. 20, No. 3, Sep. 1973, pp. 577581.
MONTGOMERY, D. C, M. S. Bazaraa and A. K. Keswani, "Inventory Models with a Mixture of
Backorders and Lost Sales," Vol. 20, No. 2, Jun. 1973, pp. 255263.
NAHMIAS, S. and W. P. Pierskalla, "Optimal Ordering Policies for a Product that Perishes in Two
Periods Subject to Stochastic Demand," Vol. 20, No. 2, Jun. 1973, pp. 207229.
NEUTS, M. F., "The Single Server Queue in Discrete TimeNumerical Analysis I," Vol. 20, No. 2,
Jun. 1973, pp. 297304.
NEUTS, M. F. and D. Heimann, "The Single Server Queue in Discrete TimeNumerical Analysis IV,"
Vol. 20, No. 4, Dec. 1973, pp. 753766.
NEUTS, M. F. and E. Klimko, "The Single Server Queue in Discrete TimeNumerical Analysis II,"
Vol. 20, No. 2, Jun. 1973, pp. 305319.
NEUTS, M. F. and E. Klimko, "The Single Server Queue in Discrete TimeNumerical Analysis III,"
Vol. 20, No. 3, Sep. 1973, pp. 557567.
810 INDEX OF VOLUME 20
NIKOLAISEN, T. and R. W. Butterworth, "Bounds on the Availability Function," Vol. 20, No. 2,
Jun. 1973, pp. 289296.
PANWALKAR, S. S., "Parametric Analysis of Linear Programs with Upper Bounded Variables,"
Vol. 20, No. 1, Mar. 1973, pp. 83~93.
PATTERSON, J. H., "Alternate Methods of Project Scheduling with Limited Resources," Vol. 20,
No. 4, Dec. 1973, pp. 767784.
PERSINGER, C. A., "Optimal Search Using Two Nonconcurrent Sensors," Vol. 20, No. 2, Jun. 1973,
pp. 277288.
PIERSKALLA, W. P. and S. Nahmias, "Optimal Ordering Policies for a Product that Perishes in Two
Periods Subject to Stochastic Demand," Vol. 20, No. 2, Jun. 1973, pp. 207229.
PRAWDA, J., "ProductionAllocation Scheduling and Capacity Expansion Using Network Flows
under Uncertainty," Vol. 20, No. 3, Sep. 1973, pp. 517531.
RAHIM, M. A. and M. Ahsanullah, "Simplified Estimates of the Parameters of the Double Exponential
Distribution Based on Optimum Order Statistics from a MiddleCensored Sample," Vol. 20, No. 4,
Dec. 1973, pp. 745751.
RAO, M. R., R. Jagannathan and W. Dent, "Parametric Linear Programming: Some Special Cases,"
Vol. 20, No. 4, Dec. 1973, pp. 725728.
RAVINDRAN, A., "A Comparison of the PrimalSimplex and Complementary Pivot Methods for
Linear Programming," Vol. 20, No. 1, Mar. 1973, pp. 95100.
ROBILLARD, P., M. Florian and P. Bratley, "On Sequencing with Earliest Starts and Due Dates
with Application to Computing Bounds for the (n/m/G/F max ) Problem," Vol. 20, No. 1, Mar. 1973,
pp. 5767.
ROGERS, W. F. and G. F. Brown, "A Bayesian Approach to Demand Estimation and Inventory Pro
visioning," Vol. 20, No. 4, Dec. 1973, pp. 607624.
ROSE, M., "Determination of the Optimal Investment in End Products and Repair Resources," Vol.
20, No. 1, Mar. 1973, pp. 147159.
RUTEMILLER, H. C. and G. G. Brown, "A Cost Analysis of Sampling Inspection under Military
Standard 105D," Vol. 20, No. 1, Mar. 1973, pp. 181199.
SCHRAGE, L. "Using Decomposition in Integer Programming," Vol. 20, No. 3, Sep. 1973, pp. 469
476.
SILVER, E. A., "Three Ways of Obtaining the Average Cost Expression in a Problem Related to
Joint Replenishment Inventory Control," Vol. 20, No. 2, Jun. 1973, pp. 241254.
SMITH, D. E., "Requirements of an 'Optimizer' for Computer Simulations," Vol. 20, No. 1, Mar.
1973, pp. 161179.
SOLAND, R. M., "An Algorithm for Separable Piecewise Convex Programming Problems," Vol. 20,
No. 2, Jun. 1973, pp. 325340.
STONE, L. D., "Total Optimality of Incrementally Optimal Allocations," Vol. 20, No. 3, Sep. 1973, pp.
419430.
TAHA, H. A., "Concave Minimization over a Convex Polyhedron," Vol. 20, No. 3, Sep. 1973, pp.
533548.
TAYLOR, J. G., "A SquaredVariable Transformation Approach to Nonlinear Programming Optimality
Conditions," Vol. 20, No. 1, Mar. 1973, pp. 25~39.
INDEX OF VOLUME 20 gjj
TAYLOR, J. G., "Target Selection in Lanchester Combat: LinearLaw Attrition Process," Vol. 20,
No. 4, Dec. 1973, pp. 673697.
THOMAS, C. J. and D. V. Mastran, "Decision Rules for Attacking Targets of Opportunity," Vol. 20,
No. 4, Dec. 1973, pp. 661672.
THOMAS, M. E. and M. Grunspan, "Hyperbolic Integer Programming," Vol. 20, No. 2, Jun. 1973,
pp. 341356.
THRALL, R. M. and D. R. Howes, "A Theory of Ideal Linear Weights for Heterogeneous Combat
Forces, Vol. 20, No. 4, Dec. 1973, pp. 645659.
TIPLITZ, C. I., "Convergence of the Bounded Fixed Charge Programming Problem," Vol. 20, No. 2,
Jun. 1973, pp. 367375.
WAGNER, H. M., "The Lower Bounded and Partial Upper Bounded Distribution Model," Vol. 20,
No. 2, Jun. 1973, pp. 265268.
WILKINSON, W. L., "Min/Max Bounds for Dynamic Network Flows," Vol. 20, No. 3, Sep. 1973, pp.
505516.
ZACKS, S. and W. J. Fenske, "Sequential Determination of Inspection Epochs for Reliability Systems
with General Life Time Distributions," Vol. 20, No. 3, Sep. 1973, pp. 377386.
CUMULATIVE INDEX FOR VOLUMES 120
ABRAMS, L. S. and B. Rasof, "A 'Static' Solution to a 'Dynamic' Problem in Acquisition Probability," Vol. 12, No. 1, Mar.
1965, pp. 6593.
AGNEW, R. A. and R. B. Hempley, "Finite Statistical Games and Linear Programming," Vol. 18, No. 1, Mar. 1971, pp. 99102.
AGNEW, R. A., "Sequential Bid Selection by Stocbastic Approximation," Vol. 19, No. 1, Mar. 1972, pp. 137143.
AGRAWA1, A. C. and I.. L. George, "Estimation of a Hidden Service Distribution of an M/CI* System," Vol. 20, No. 3, Sept.
1973, pp. 549555.
AHRENS, J. H., "Suboptimal Algorithms for the Quadratic Assignment Problem," Vol. 15, No. 1, Mar. 1968, pp. 4962.
AHSANULLAH, M. and M. A. Rahim, "Simplified Estimates of the Parameters of the Double Exponential Distribution Based
on Optimum Order Statistics from a MiddleCensored Sample," Vol. 20, No. 4, Dec. 1973, pp. 745751.
AHSANULLAH, M. and A. K. Md. E. Saleh, "Optimum Allocation of Quantiles in Disjoint Intervals for the Blues of the Param
eters of Exponential Distribution when the Sample is Censored in the Middle," Vol. 17, No. 3, Sept. 1970, pp. 331349.
ALAM, K., "Peak Rate of Occurrence of a Poisson Process," Vol. 20, No. 2, Jun. 1973, pp. 269275.
ALLEN, S. G., "Redistribution of Total Stock Over Several User Locations," Vol. 5, No. 4, Dec. 1958, pp. 337345.
ALLEN, W. R., "Simple Inventory Models with Bunched Inputs," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 265273.
ALMOGY, Y. and O. Levin, "The Fractional FixedCharge Problem," Vol. 18, No. 3, Sept. 1971, pp. 307315.
ALTER, R. and B. P. Lientz, "Applications of a Generalized Combinatorial Problem of Smirnov," Vol. 16, No. 4, Dec. 1969,
pp. 543547.
ALTER, R. and B. Lientz, "A Note on a Problem of Smirnov: A Graph Theoretic Interpretation," Vol. 17, No. 3, Sept. 1970,
pp. 407408.
ALWAY, G. G., "A Triangularization Method for Computations in Linear Programming," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962,
pp. 163180.
ANCKER, C. J., JR. and A. V. Gafarian, "Queueing with Reneging and Multiple Heterogeneous Servers," Vol. 10, No. 2, June
1963, pp. 125149.
ANCKER, C. J., JR. and A. V. Gafarian, "The Distribution of Rounds Fired in Stochastic Duels," Vol. 11, No. 4, Dec. 1964,
pp. 303327.
ANCKER, C. J.. JR. and A. V. Gafarian, "The Distribution of the TimeDuration of Stochastic Duels," Vol. 12, Nos. 3 & 4,
Sept.Dec. 1965, pp. 275294.
ANDERSON, B., "A New Field for Logistics Research," Vol. 1, No. 2, June 1954, pp. 7981.
ANDREWS, R. A., "A Note on a Role of Intelligence Analysis in a Logistics Operations Research," Vol. 10, No. 2, June 1963,
pp. 193195.
ANTELMAN, G. and I. R. Savage, "Characteristic Functions of Stochastic Integrals and Reliability Theory," Vol. 12, Nos. 3
&4, Sept.Dec. 1965, pp. 199222.
ANTELMAN, G. and I. R. Savage, "Surveillance Problems: Wiener Processes," Vol. 12, No. 1, Mar. 1965, pp. 3555.
ANTOSIEWICZ, H. A., "Analytic Study of War Games," Vol. 2, No. 3, Sept. 1955, pp. 181208.
APPLE, R. E. and D. E. Farrar, "Some Factors That Affect the Overhaul Cost of Ships: An Exercise in Statistical Cost Analysis,"
Vol. 10, No. 4, Dec. 1963, pp. 335368.
ARORA, K. L., "A Generalized Problem in Air Defense," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 281287.
ARORA, K. L. and C. Mohan, "Analytical Study of a Problem in Air Defense," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 275~279.
ARROW, K. J., L. Hurwicz, and H. Uzawa, "Constraint Qualifications in Maximization Problems," Vol. 8, No. 2, June 1961,
pp. 175191.
ARTHANARI, T. S. and A. C. Mukhopadhyay, "A Note on a Paper by W. Szwarc," Vol. 18, No. 1, Mar. 1971, pp. 135138.
ATWATER, T. V. V., JR., "The Theory of Inventory Management A Review," Vol. 1, No. 4, Dec. 1954, pp. 295300.
AUMANN, R. J., "Linearity of Unrestrictedly Transferable Utilities," Vol. 7, No. 3, Sept. 1960, pp. 281284.
AUMANN, R. J. and J. B. Kruskal, "Assigning Quantitative Values to Qualitative Factors in the Naval Electronics Problem,"
Vol. 6, No. 1, Mar. 1959, pp. 116.
AUMANN, R. J. and J. B. Kruskal, "The Coefficients in an Allocation Problem," Vol. 5, No. 2, June 1958, pp. 111123.
BAKER, K. R., and A. G. Merten, "Scheduling with Parallel Processors and Linear Delay Costs," Vol. 20, No. 4, Dec. 1973,
pp. 793804.
BALAS, E., "Machine Sequencing: Disjunctive Graphs and DegreeConstrained Subgraphs," Vol. 17, No. 1, Mar. 1970, pp.
110.
BALINSKI, M. L., "FixedCost Transportation Problems," Vol. 8, No. 1, Mar. 1961, pp. 4154.
BARANKIN, E. W., "A DeliveryLag Inventory Model With an Emergency Provision (The SinglePeriod Case)," Vol. 8, No. 3,
Sept. 1961, pp. 285311.
BARON, D. P., "Quadratic Programming with Quadratic Constraints," Vol. 19, No. 2, June 1972, pp. 253260.
BARON, D. P., "Information in TwoStage Programming Under Uncertainty," Vol. 18, No. 2, June 1971, pp. 169176.
813
814 CUMULATIVE INDEX FOR VOLUMES 120
BARR, D. R., W. E. Coleman and T. Jayachandran, "A Note on a Comparison of Confidence Interval Techniques in Truncated
Life Tests," Vol. 18, No. 4, Dec. 1971, pp. 567569.
BARTLETT, T. E., "An Algorithm for the Minimum Units Required to Maintain a Specified Schedule," Vol. 4, No. 2, June
1957, pp. 139149.
BARTLETT, T. E. and A. Charnes, "Cyclic Scheduling and Combinatorial Topology; Assignment and Routing of Motive Power
to Meet Scheduling and Maintenance Requirements," Part II "Generalization and Analysis," Vol. 4, No. 3, Sept. 1957,
pp. 207220.
BASU, A. P., "On a Sequential Rule for Estimating the Location Parameter of an Exponential Distribution," Vol. 18, No. 3,
Sept. 1971, pp. 329337.
BAUMOL, W. J. and H. W. Kuhn, "An Approximative Algorithm for the FixedCharges Transportation Problem," Vol. 9, No.
1, Mar. 1962, pp. 115.
BAZARAA, M. S., "Geometry and the Resolution of Duality Gaps," Vol. 20, No. 2, June 1973, pp. 357366.
BAZARAA, M. S., D. C. Montgomery and A. K. Keswani, "Inventory Models with a Mixture of Backorders and Lost Sales,"
Vol. 20, No. 2, June 1973, pp. 255263.
BEALE, E. M. L., "An Algorithm for Solving the Transportation Problem When the Shipping Cost Over Each Route is Con
vex," Vol. 6, No. 1, Mar. 1959, pp. 4356.
BEALE, E. M. L., "Cycling in the Dual Simplex Algorithm," Vol. 2, No. 4, Dec. 1955, pp. 269275.
BEALE, E. M. L., "On Quadratic Programming," Vol. 6, No. 3, Sept. 1959, pp. 227243.
BEALE, E. M. L. and G. P. M. Heselden, "An Approximate Method of Solving Blotto Games," Vol. 9, No. 2, June 1962, pp.
6579.
BECKMANN, M. J., "An Inventory Policy for Repair Parts," Vol. 6, No. 3, Sept. 1959, pp. 209220.
BECKMANN, M. J. and F. Bobkoski, "Airline Demand: An Analysis of Some Frequency Distributions," Vol. 5, No. 1, Mar.
1958, pp. 4351.
BECKMANN, M. J. and J. Laderman, "A Bound on the Use of Inefficient Indivisible Units," Vol. 3, No. 4, Dec. 1956, pp. 245252.
BEGEDDOV, A. G., "Contract Award Analysis by Mathematical Programming," Vol. 17, No. 3, Sept. 1970, pp. 297307.
BELL, C. E., "Multiple Dispatches in a Poisson Process," Vol. 17, No. 1, Mar. 1970, pp. 99102.
BELLMAN, R., "Decision Making in the Face of Uncertainty I," Vol. 1, No. 3, Sept. 1954, pp. 230232.
BELLMAN, R., "Decision Making in the Face of Uncertainty II," Vol. 1, No. 4, Dec. 1954, pp. 327332.
BELLMAN, R., "Formulation of Recurrence Equations for Shuttle Process and Assembly Line," Vol. 4, No. 4, Dec. 1957, pp.
321334.
BELLMAN, R., "Notes on the Theory of Dynamic Programming — IV — Maximization Over Discrete Sets," Vol. 3, Nos. 1 & 2,
MarJune 1956, pp. 6770.
BELLMAN, R., "On Some Applications of the Theory of Dynamic Programming," Vol. 1, No. 2, June 1954, pp. 141153.
BELLMAN, R. and S. Dreyfus, "A Bottleneck Situation Involving Interdependent Industries," Vol. 5, No. 4, Dec. 1958, pp.
307314.
BELLMORE, M., "A Maximum Utility Solution to a Vehicle Constrained Tanker Scheduling Problem," Vol. 15, No. 3, Sept.
1968, pp. 403412.
BELLMORE, M., J. C. Liebman and D. H. Marks, "An Extension of the (Szwarc) Truck Assignment Problem," Vol. 19, No. 1,
Mar. 1972, pp. 9199.
BELLMORE, M., W. D. Eklof and G. L. Nemhauser, "A Decomposable Transshipment Algorithm for a Multiperiod Transpor
tation Problem," Vol. 16, No. 4, Dec. 1969, pp. 517524.
BELLMORE, M., G. Bennington and S. Lubore, "A Network Isolation Algorithm," Vol. 17, No. 4, Dec. 1970, pp. 461469.
BELON, R. G., Logistics Research Programs of the U.S. Army, U.S. Air Force, and U.S. Navy. "Briefing on Army Logistics
Research," Vol. 5, No. 3, Sept. 1958, pp. 221223.
BENNETT, G. K. and H. F. Martz, "An Empirical Bayes Estimator for the Scale Parameter of the TwoParameter Weibull
Distribution," Vol. 20, No. 3, Sept. 1973, pp. 387393.
BENNINGTON, G., "A Maximum Utility Solution to a Vehicle Constrained Tanker Scheduling Problem," Vol. 15, No. 3, Sept.
1968, pp. 403412.
BENNINGTON, G., L. Gsellman and S. Lubore, "An Economic Model for Planning Strategic Mobility Posture," Vol. 19, No. 3,
Sept. 1972, pp. 461470.
BENNINGTON, G., M. Bellmore and S. Lubore, "A Network Isolation Algorithm," Vol. 17, No. 4, Dec. 1970, pp. 461469.
BENNINGTON, G., and S. Lubore, "Resource Allocation for Transportation," Vol. 17, No. 4, Dec. 1970, pp. 471484.
BERGER, P. D., "Prediction with ZeroOne Loss Structure," Vol. 19, No. 1, Mar. 1972, pp. 159164.
BERRY, S. D., "Economic Impact and the Notion of Compensated Procurement," Vol. 15, No. 1, Mar. 1968, pp. 6379.
BESSLER, S. A., "An Application of Servomechanisms to Inventory," Vol. 15, No. 2, June 1968, pp. 157168.
BESSLER, S. A. and A. F. Veinott, Jr., "Optimal Policy for a Dynamic MultiEchelon Inventory Model," Vol. 13, No. 4, Dec.
1966, pp. 355389.
BEUTLER, F. J. and O. A. Z. Leneman, "On a New Approach to the Analysis of Stationary Inventory Problems," Vol. 16, No.
1, Mar. 1969, pp. 115.
CUMULATIVE INDEX FOR VOLUMES 120 gl5
BHASHYAM, N., "Stochastic Duels with Letal Dose," Vol. 17, No. 3, Sept. 1970, pp. 397405.
BHASHYAM, N., "Stochastic Duels with Nonrepairable Weapons," Vol. 17, No. 1, Mar. 1970, pp. 121129.
BILES, W. E. and J. W. Schmidt, Jr., "A Note on a Paper by Houston and Huffman," Vol. 19, No. 3. Sept. 1972, pp. 561567.
BLACHMAN, N. M., "Prolegomena to Optimum Discrete Search Procedures," Vol. 6, No. 4, Dec. 1959, pp. 273281.
BLACKETT, D. W., "Pure Strategy Solutions of Blotto Games," Vol. 5, No. 2, June 1958, pp. 107109.
BLACKETT, D. W., "Some Blotto Games," Vol. 1, No. 1, Mar. 1954, pp. 5560.
BLACKWELL, D., "On MultiComponent Attrition Games," Vol. 1, No. 3, Sept. 1954, pp. 210216.
BLITZ, M., "Optimum Allocation of a Spares Budget," Vol. 10, No. 2, June 1963, pp. 175191.
BLUMENTHAL, S., "Interval Estimation of the Normal Mean Subject to Restrictions, When the Variance Is Known," Vol.
17, No. 4, Dec. 1970, pp. 485505.
BOBKOSKI, F. and M. J. Beckmann, "Airline Demand: An Analysis of Some Frequency Distributions," Vol. 5, No. 1, Mar.
1958, pp. 4351.
BOL, G. and O. Moeschlin, "Applications of Mills' Differential," Vol. 20, No. 1, Mar. 1973, pp. 101108.
BOLL, C, "Cannibalization in Multicomponent Systems and the Theory of Reliability," Vol. 15, No. 3, Sept. 1968, pp. 331360.
BONESS, A. J. and A. N. Schwartz, "Aircraft Replacement Policies in the Naval Advanced Jet Pilot Training Program: A
Practical Example of DecisionMaking Under Incomplete Information," Vol. 16, No. 2, June 1969, pp. 237257.
BOOK, S. A.. "Large Deviation Probabilities for Order Statistics," Vol. 18, No. 4, Dec. 1971, pp. 521523.
BOVAIRD, R. L., and H. I. Zagor, "Lognormal Distribution and Maintainability in Support Systems Research," Vol. 8, No. 4,
Dec. 1961, pp. 343356.
BOWMAN, V. J., JR. and G. L. Nemhauser, "A Finiteness Proof for Modified Dantzig Cuts in Integer Programming," Vol. 17,
No. 3, Sept. 1970, pp. 309313.
BRACKEN, J., "A Linear Programming Model for MinimumCost Procurement and Operation of Marine Corps Training Air
craft," Vol. 15, No. 1, Mar. 1968, pp. 8197.
BRACKEN, J., "Programming the Procurement of Airlift and Sealift Forces: A Linear Programming Model for Analysis of
the LeastCost Mix of Strategic Deployment Systems," Vol. 14, No. 2, June 1967, pp. 241255.
BRACKEN, J., C. B. Brossman, C. B. Magruder, and A. D. Tholen, "A Theater Materiel Model," Vol. 12, Nos. 3 & 4, Sept.
Dec. 1965, pp. 295313.
BRACKEN, J. and J. D. Longhill, "Note on a Model for Minimizing," Vol. 1 1 , No. 4, Dec. 1964, pp. 359364.
BRACKEN, J. and K. W. Simmons, "Minimizing Reductions in Readiness Caused by Time Phased Decreases in Aircraft Over
haul and Repair Activities," Vol. 13, No. 2, June 1966, pp. 159165.
BRACKEN, J. and R. M. Soland, "Statistical Decision Analysis of Stochastic Linear Programming Problems," Vol. 13, No. 3,
Sept. 1966, pp. 205225.
BRACKEN, J., E. W. Rice and A. W. Pennington, "Allocation of CarrierBased Attack Aircraft Using NonLinear Program
ming," Vol. 18, No. 3, Sept. 1971, pp. 379393.
BRACKEN, J. and T. C. Varley, "A Model for Determining Protection Levels for Equipment Classes within a Set of Subsys
tems," Vol. 10, No. 3, Sept. 1963, pp. 257262.
BRAITHWAITE, R. B., "A Terminating Iterative Algorithm for Solving Certain Games and Related Sets of Linear Equations,"
Vol. 6, No. 1, Mar. 1959, pp. 6374.
BRANDENBURG, R. G. and A. C. Stedry, "Toward a MultiStage Information Conversion Model of the Research and Develop
ment Process," Vol. 13, No. 2, June 1966, pp. 129146.
BRANDT, E. B. and D. R. Limaye, "MAD: Mathematical Analysis of Downtime," Vol. 17, No. 4, Dec. 1970, pp. 525534.
BRATLEY, P., M. Florian and P. Robillard, "Scheduling with Earliest Start and Due Date Constraints," Vol. 18, No. 4, Dec.
1971, pp. 511519.
BRATLEY, P., M. Florian and P. Robillard, "On Sequencing with Earliest Starts and Due Dates with Application to Computing
Bounds for the (n/m/C/F max ) Problem," Vol. 20, No. 1, Mar. 1973, pp. 5767.
BRECHT, H. D. and A. C. Stedry, "Toward Optimal Bidding Strategies," Vol. 19, No. 3, Sept. 1972, pp. 423434.
BREIMAN, L., "Investment Policies for Expanding Businesses Optimal in a LongRun Sense," Vol. 7, No. 4, Dec. 1960, pp.
647651.
BREINING, P. and H. M. Salkin, "Integer Points on the Gomory Fractional Cut (Hyperplane)," Vol. 18, No. 4, Dec. 1971, pp.
491496.
BREMER, H., W. Hall, and M. Paulsen, "Experiences With the Bid Evaluation Problem (Abstract)," Vol. 4, No. 1, Mar. 1957,
p. 27.
BRENNER, J. L., "Stock Control in a Many Depot System," Vol. 16, No. 3, Sept. 1969, pp. 359379.
BREUER, M. A., "The Formulation of Some Allocation and Connection Problems As Integer Problems," Vol. 13, No. 1, Mar.
1966, pp. 8395.
BRIGGS, F. E. A., "Solution of the Hitchcock Problem with One Single Row Capacity Constraint per Row by the FordFulkerson
Method," Vol. 9, No. 2, June 1962, pp. 107120.
BROCK, P., W. D. Correl, and G. W. Evans, II, "Techniques for Evaluating Military Organizations and Their Equipment,"
Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 211229.
BRODSKY, N., "The Need for a Strategy of Logistics Research," Vol. 7, No. 4, Dec. 1960, pp. 295297.
816 CUMULATIVE INDEX FOR VOLUMES 120
BROOKS, R. B. S. and J. Y. Lu, "War Reserve Spares Kits Supplemented by Normal Operating Assets," Vol. 16, No. 2, June
1969, pp. 229236.
BROSSMAN, C. B., J. Bracken, C. B. Magruder, and A. D. Tholen, "A Theater Materiel Model," Vol. 12, Nos. 3 & 4, Sept.
Dec. 1965, pp. 295313.
BROWN, B., "A Comparative Study of Prediction Techniques," Vol. 7, No. 4, Dec. 1960, pp. 471492.
BROWN, G. F. and W. F. Rogers, "A Bayesian Approach to Demand Estimation and Inventory Provisioning," Vol. 20, No. 4,
Dec. 1973, pp. 607624.
BROWN, G. G. and H. C. Rutemiller, "A Cost Analysis of Sampling Inspection under Military Standard 105D," Vol. 20, No. 1,
Mar. 1973, pp. 181199.
BROWN, R. G., "Estimating Aggregate Inventory Standards," Vol. 10, No. 1, Mar. 1963, pp. 5571.
BROWN, R. G., "Simulations to Explore Alternative Sequencing Rules," Vol. 15, No. 2, June 1968, pp. 281286.
BROWN, R. G. and G. Gerson, "Decision Rules for Equal Shortage Policies," Vol. 17, No. 3, Sept. 1970, pp. 351358.
BROWN, R. H., Logistics Research Programs of the U.S. Army, U.S. Air Force, and U.S. Navy. "Briefing on the Logistics
Research Program of the Air Force," Vol. 5, No. 3, Sept. 1958, pp. 223225.
BRYAN, J. G., G. P. Wadsworth, and T. M. Whitin, "A MultiStage Inventory Model," Vol. 2, Nos. 1 & 2, Mar.June 1955, pp.
2537.
BURDET, C. "Polaroids: A New Tool in NonConvex and in Integer Programming," Vol. 20, No. 1, Mar. 1973, pp. 1324.
BURNHAM, P. R., "A Linear Programming Model for MinimumCost Procurement and Operation of Marine Corps Training
Aircraft," Vol. 15, No. 1, Mar. 1968, pp. 8197.
BURT, J. M., JR., D. P. Gaver and M. Perlas, "Simple Stochastic Networks: Some Problems and Procedures," Vol. 17, No. 4,
Dec. 1970, pp. 439459.
BURTON, R. W. and S. C. Jaquette, "The Initial Provisioning Decision for Insurance Type Items," Vol. 20, No. 1, Mar. 1973,
pp. 123146.
BUSBY, J. C, "Comments on the Morgenstern Model," Vol. 2, No. 4, Dec. 1955, pp. 225236.
BUTTERWORTH, R. W. and T. Nikolaisen, "Bounds on the Availability Function," Vol. 20, No. 2, June 1973, pp. 289296.
BUZACOTT, J. A. and S. K. Dutta, "Sequencing Many Jobs on a MultiPurpose Facility," Vol. 18, No. 1, Mar. 1971, pp. 7582.
CABOT, A. V. and R. L. Francis. "Properties of a Multifacility Location Problem Involving Euclidian Distances," Vol. 19,
No. 2, Jun. 1972, pp. 335353.
CALDWELL, W. V., C. H. Coombs, M. S. Schoeffler, and R. M. Thrall, "A Model for Evaluating the Output of Intelligence
Systems," Vol. 8, No. 1, Mar. 1961, pp. 2540.
CALDWELL, W. V., R. M. Thrall, and C. H. Coombs, "Linear Model for Evaluating Complex Systems," Vol. 5, No. 4, Dec.
1958, pp. 347361.
CALLAHAN, J. R., "A Queue with Waiting Time Dependent Service Times," Vol. 20, No. 2, June 1973, pp. 321324.
CAMPBELL, R. D., F. D. Dorey, and R. E. Murphy, "Concept of a Logistics System," Vol. 4, No. 2, June 1957, pp. 101116.
CANDLER, W. and R. J. Townsley, "Quadratic as Parametric Linear Programming," Vol. 19, No. 1, Mar. 1972, pp. 183189.
CANNON, E. W., "The Reflection of Logistics in Electronic Computer Design," Vol. 7, No. 4, Dec. 1960, pp. 365371.
CARNEY, R. B., "Some General Observations and Experiences in Logistics," Vol. 3, Nos. 1 & 2, Mar.June 1956, pp. 19.
CASSIDY, R. G., C. A. Field, and M. J. L. Kirby, "Partial Information in Two Person Games with Random Payoffs," Vol. 20,
No. 1, Mar. 1973, pp. 4156.
CHAN, L. K., "Linear Estimation of the Location and Scale Parameters from Type II Censured Samples from Symmetric
Unimodal Distributions," Vol. 14, No. 2, June 1967, pp. 135145.
CHAN, L. K. and A. B. M. Lutful Kabir, "Optimum Quantiles for the Linear Estimation of the Parameters of the Extreme Value
Distribution in Complete and Censored Samples," Vol. 16, No. 3, Sept. 1969, pp. 381404.
CHAN, L. K., S. W. H. Cheng and E. R. Mead, "An Optimum tTest for the Scale Parameter of an ExtremeValue Distribution,"
Vol. 19, No. 4, Dec. 1972, pp. 715723.
CHANDRASEKARAN, R., and K. P. K. Nair, "Optimal Location of a Single Service Center of Certain Types," Vol. 18, No.
4, Dec. 1971, pp. 503510.
CHANG, R. C. and S. Ehrenfeld, "On a Sequential Test Procedure with Delayed Observations," Vol. 19, No. 4, Dec. 1972,
pp. 651661.
CHANG, W., "Congestion Analysis of a Computer Core Storage System," Vol. 14, No. 3, Sept. 1967, pp. 367379.
CHARNES, A., "On Some Stochastic Tactical Antisubmarine Games," Vol. 14, No. 3, Sept. 1967, pp. 291311.
CHARNES, A. "Structural Sensitivity Analysis in Linear Programming and in Exact Product Form Left Inverse," Vol. 15,
No. 4, Dec. 1968. pp. 517522.
CHARNES, A. and T. E. Bartlett, "Cyclic Scheduling and Combinatorial Topology: Assignment and Routing of Motive Power
to meet Scheduling and Maintenance Requirements." Part II, "Generalization and Analysis," Vol. 4, No. 3, Sept. 1957,
pp. 207220.
CHARNES, A. and W. W. Cooper, "Nonlinear Network Flows and Convex Programming Over Incidence Matrices," Vol. 5,
No. 3, Sept. 1958, pp. 231240.
CHARNES, A. and W. W. Cooper, "Programming with Linear Fractional Functionals," Vol. 9, Nos. 3 & 4, Sept. Dec. 1962,
pp. 181186.
CUMULATIVE INDEX FOR VOLUMES 120 817
CHARNES, A. and W. W. Cooper, "Some Problems and Models for TimePhased Transport Requirements," Vol. 7, No. 4,
Dec. 1960, pp. 533544.
CHARNES, A., W. W. Cooper, and M. Miller, "Dyadic Programs and Subdual Methods," Vol. 8, No. 1, Mar. 1961, pp. 123.
CHARNES, A. and C. E. Lemke, "Minimization of NonLinear Separable Convex Functionals," Vol. 1, No. 4, Dec. 1954, pp.
301312.
CHARNES, A. and M. H. Miller, "Mathematical Programming and Evaluation of Freight Shipment Systems, Application and
Analysis," Part II, "Analysis," Vol. 4, No. 3, Sept. 1957, pp. 243252.
CHARNES, A., W. W. Cooper and K. O. Kortanek, "On the Theory of SemiInfinite Programming and a Generalization of the
KuhnTucker Saddle Point Theorem for Arbitrary Convex Functions," Vol. 16, No. 1, Mar. 1969, pp. 4151.
CHARNES, A., F. Glover and D. Klingman, "The Lower Bounded and Partial Upper Bounded Distribution Model," Vol. 18,
No. 2, Jun. 1971, pp. 277281.
CHARNES, A. and W. W. Cooper, "An Explicit General Solution in Linear Fractional Programming," Vol. 20, No. 3, Sep.
1973, pp. 449467.
CHATTOPADHYAY, R., "Differential Game Theoretic Analysis of a Problem of Warfare," Vol. 16, No. 3, Sept. 1969, pp.
435441.
CHAUDHRY, M. L., "On the DiscreteTime Queue Length Distribution under MarkovDependent Phases," Vol. 19, No. 2,
Jun. 1972, pp. 369378.
CHENEV, L. K., "Linear Program Planning of Refinery Operations," Vol. 4, No. 1, Mar. 1957, pp. 916.
CHENG, S. W. H., L. K. Chan and E. R. Mead. "An Optimum TTest for the Scale Parameter of an ExtremeValue Distribution,"
Vol. 19, No. 4, Dec. 1972, pp. 715723.
CHOE, U. C. and D. A. Schrady, "Models for MultiItem Continuous Review Inventory Policies Subject to Constraints," Vol.
18, No. 4, Dec. 1971, pp. 451463.
CLARK, A. J., "The Use of Simulation to Evaluate a Multiechelon, Dynamic Inventory Model," Vol. 7, No. 4, Dec. 1960, pp.
429445.
CLARK, A. J., "An Informal Survey of MultiEchelon Inventory Theory," Vol. 19, No. 4, Dec. 1972, pp. 621650.
COFFMAN, E. G., JR., "Markov Chain Analyses of Multiprogrammed Computer Systems," Vol. 16, No. 2, June 1969, pp. 175
197.
COFFMAN, E. G., JR., "Bounds on ParallelProcessing of Queues with MultiplePhase Jobs," Vol. 14, No. 3, Sept. 1967, pp.
345366.
COHEN, E. A., JR. and K. D. Shere, "A Defense Allocation Problem with Development Costs," Vol. 19, No. 3, Sep. 1972, pp.
525537.
COHEN, N. D., "An AttackDefense Game with Matrix Strategies," Vol. 13, No. 4, Dec. 1966, pp. 391402.
COLEMAN, W. E., T. Jayachandran and D. R. Barr, "A Note on a Comparison of Confidence Interval Techniques in Trun
cated Life Tests," Vol. 18, No. 4, Dec. 1971, pp. 567569.
COLLINS, F. R., JR. and D. Guthrie, Jr., "A Model for the Analysis of AEW and CAP Aircraft Availability," Vol. 10, No. 1,
Mar. 1963, pp. 7379.
COOK, F. X., A. S. Rhode and J. J. Gelke, "Impact of an All Volunteer Force upon the Navy in the 19721973 Timeframe,"
Vol. 19, No. 1, Mar. 1972, pp. 4375.
COOMBS, C. H., W. V. Caldwell, M. S. Schoeffler, and R. M. Thrall, "A Model for Evaluating the Output of Intelligence Sys
tems," Vol. 8, No. 1, Mar. 1961, pp. 2540.
COOMBS, C. H., W. V. Caldwell, and R. M. Thrall, "Linear Model for Evaluating Complex Systems," Vol. 5, No. 4, Dec. 1958,
pp. 347361.
COOPER, C. R., J. A. Sheler and A. N. Schwartz, "Dynamic Programming Approach to the Optimization of Naval Aircraft
Rework and Replacement Policies," Vol. 18, No. 3, Sept. 1971, pp. 395414.
COOPER, L., "An Approximate Solution Method for the Fixed Charge Problem," Vol. 14, No. 1, Mar. 1967, pp. 101113.
COOPER, W. W., "Structural Sensitivity Analysis in Linear Programming and An Exact Product Form Left Inverse," Vol.
15, No. 4, Dec. 1968, pp. 517522.
COOPER, W. W. and A. Charnes, "Nonlinear Network Flows and Convex Programming Over Incidence Matrices," Vol. 5,
No. 3, Sept. 1958, pp. 231240.
COOPER, W. W. and A. Charnes, "Programming with Linear Fractional Functionals," Vol. 9, Nos 3 & 4, Sept.Dec. 1962,
pp. 181186.
COOPER, W. W. and A. Charnes, "Some Problems and Models for TimePhased Transport Requirements," Vol. 7, No. 4,
Dec. 1960, pp. 533544.
COOPER, W. W., A. Charnes, and M. Miller, "Dyadic Programs and Subdual Methods," Vol. 8, No. 1, Mar. 1961, pp. 123.
COOPER, W. W., A Charnes and K. O. Kortanek, "On the Theory of SemiInfinite Programming and a Generalization of the
KuhnTucker Saddle Point Theorem for Arbitrary Convex Functions," Vol. 16, No. 1, Mar. 1969, pp. 4151.
COOPER, W. W. and A. Charnes, "An Explicit General Solution in Linear Fractional Programming," Vol. 20, No. 3, Sep. 1973,
pp. 449467.
CORRELL, W. D., P. Brock, and G. W. Evans, II, "Techniques for Evaluating Military Organizations and Their Equipment,"
Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 211229.
COZZOLINO, J. M., "Probabilistic Models of Decreasing Failure Rate Processes," Vol. 15, No. 3, Sept. 1968, pp. 361374.
818 CUMULATIVE INDEX FOR VOLUMES 120
COZZOLINO, J. M., "The Optimal BurnIn Testing of Repairable Equipment," Vol. 17, No. 2, June 1970, pp. 167181.
CRABILL, T. B. and W. L. Maxwell, "Single Machine Sequencing with Random Processing Times and Random DueDates,"
Vol. 16, No. 4, Dec. 1969, pp. 549554.
CRANE, R. R., "Some Recent Developments in Transportation Research," Vol. 4, No. 3, Sept. 1957, pp. 173181.
CRAVEN, B. D. and B. Mond, "A Note on Mathematical Programming with Fractional Objective Functions," Vol. 20, No. 3,
Sep. 1973, pp. 577581.
CREMEANS, J. E., R. A. Smith, and G. R. Tyndall, "Optimal Multicommodity Network Flows with Resource Allocation,"
Vol. 17, No. 3, Sept. 1970, pp. 269279.
CREMEANS, J. E. and H. S. Weigel, "The Multicommodity Network Flow Model Revised to Include Vehicle Per time Period
and Node Constraints," Vol. 19, No. 1, Mar. 1972, pp. 7789.
CROSS, J. G., "On Professor Schelling's Strategy of Conflict," Vol. 8, No. 4, Dec. 1961, pp. 421425.
CROW, R. T., "An Approach to the Allocation of Common Costs of MultiMission Systems," Vol. 20, No. 3, Sep. 1973, pp.
431447.
CROWSTON, W. B. and J. F. Pierce, "TreeSearch Algorithms for Quadratic Assignment Problems," Vol. 18, No. 1, Mar.
1971, pp. 136.
CUNNINGHAM, A. A. and A. L. Saipe. "Heuristic Solution to a Discrete Collection Model," Vol. 19, No. 2, Jun. 1972, pp.
379388.
CUNNINGHAM, A. A. and S. K. Dutta, "Scheduling Jobs, with Exponentially Distributed Processing Times, on Two Machines
of a Flow Shop," Vol. 20, No. 1, Mar. 1973, pp. 6981.
CURTIS, I. N., "Logistics Without Storage," Vol. 2, No. 3, Sept. 1955, pp. 125128.
DANSKIN, J. M., "A Game Over Spaces of Probability Distribution," Vol. 11, Nos. 2 and 3, JuneSept. 1964, pp. 157189.
DANSKIN, J. M., "Fictitious Play for Continuous Games," Vol. 1, No. 4, Dec. 1954, pp. 313320.
DANSKIN, J. M., "Mathematical Treatment of a Stockpiling Problem," Vol. 2, Nos. 1 & 2, Mar. June 1955, pp. 99109.
DANTZIG. G. B., "Note on Solving Linear Programs in Integers," Vol. 6, No. 1, Mar. 1959, pp. 7576.
DANTZIG. G. B., "The Fixed Charge Problem," Vol. 15, No. 3, Sept. 1968. pp. 413424.
DANTZIG, G. B. and D. R. Fulkerson, "Computation of Maximal Flows in Networks," Vol. 2, No. 4, Dec. 1955, pp. 277283.
DANTZIG. G. B. and D. R. Fulkerson, "Minimizing the Number of Tankers to Meet a Fixed Schedule," Vol. 1, No. 3, Sept. 1954,
pp. 217222.
DAS, P., "Effect of SwitchOver Devices on Reliability of a Standby Complex System," Vol. 19, No. 3, Sept. 1972, pp. 517523.
DAUBIN, S. C, "The Allocation of Development Funds: An Analytic Approach," Vol. 5, No. 3, Sept. 1958, pp. 263276.
DAVID, H., "The BuildUp Time of Waiting Lines," Vol. 7, No. 2, June 1960, pp. 185193.
DAVIS, H., "A Mathematical Evaluation of a Work Sampling Technique," Vol. 2, Nos. 1 & 2, Mar.June 1955, pp. 111117.
DAVIS, M. and M. Maschler, "The Kernel of a Cooperative Game," Vol. 12, Nos. 3 & 4, Sept.Dec. 1965, pp. 223259.
DAVIS, P. L., and T. L. Ray, "A BranchBound Algorithm for the Capacitated Facilities Location Problem," Vol. 16, No. 3,
Sept. 1969, pp. 331334.
DAY, J. E. and M. P. Hottenstein, "Review of Sequencing Research," Vol. 17, No. 1, Mar. 1970, pp. 1139.
DEE, N. and J. C. Liebman, "Optimal Location of Public Facilities," Vol. 19, No. 4, Dec. 1972, pp. 753759.
DELBROUCK, L. E. N., "The Law of Averages as a Computing Tool," Vol. 19, No. 1, Mar. 1972, pp. 149158.
DELFAUSSE, J. and S. Saltzman, "Values for Optimum Reject Allowance," Vol. 13, No. 2, June 1966, pp. 147157.
DELLINGER. D. C, "On Some Economic Concepts of Multiple Incentive Contracting," Vol. 15, No. 4, Dec. 1968, pp. 477489.
DELLINGER, D. C, "An Application of Linear Programming to Contingency Planning: A Tactical Airlift System Analysis,"
Vol. 18, No. 3, Sept. 1971, pp. 357378.
DENICOFF, M., J. Fennell, S. E. Haber, W. H. Marlow, F. W. Segel, and H. Solomon, "The Polaris Military Essentiality Sys
tem," Vol. 11, No. 4, Dec. 1964, pp. 235257.
DENICOFF, M., J. Fennell, S. E. Haber, W. H. Marlow, and H. Solomon, "A Polaris Logistics Model," Vol. 11, No. 4, Dec. 1964,
pp. 259272.
DENICOFF, M., J. P. Fennell, and H. Solomon, "Summary of a Method for Determining the Military Worth of Spare Parts,"
Vol. 7, No. 3, Sept. 1960, pp. 221234.
DENICOFF, M. and H. Solomon, "Simulations of Alternative Allowance List Policies," Vol. 7, No. 2, June 1960, pp. 137149.
DENT, W., R. Jagannathan and M. R. Rao, "Parametric Linear Programming: Some Special Cases," Vol. 20, No. 4, Dec. 1973,
pp. 725728.
DENZLER, D. R.. "An Approximative Algorithm for the Fixed Charge Problem," Vol. 16, No. 3, Sept. 1969, pp. 411416.
DERMAN, C, G. J. Lieberman and S. M. Ross, "On Optimal Assembly of Systems," Vol. 19, No. 4, Dec. 1972, pp. 569574.
DERMAN, C, "On Minimax Surveillance Schedules," Vol. 8, No. 4, Dec. 1961, pp. 415419.
DERMAN, C. and J. Sacks, "Replacement of Periodically Inspected Equipment," Vol. 7, No. 4, Dec. 1960, pp. 597607.
DERMAN. C. and M. Klein, "Surveillance of MultiComponent Systems: A Stochastic Traveling Salesman's Problem," Vol. 13,
No. 2, June 1966, pp. 103111.
D'ESOPO, D. A., "A Convex Programming Procedure," Vol. 6, No. 1, Mar. 1959, pp. 3242.
D'ESOPO, D. A., H. L. Dixion, and B. Lefkowitz, "A Model for Simulating an AirTransportation System," Vol. 7, No. 3, Sept.
1960, pp. 213220.
CUMULATIVE INDEX FOR VOLUMES 120 819
D'ESOPO, D. A., and B. Lefkowitz, "Note on an Integer Linear Programming Model for Determining a Minimum Embarkation
Fleet," Vol. 11, No. 1, Mar. 1964, pp. 7982.
DEVANNEY, J. W. Ill, "A Note on Adaptive Boiler Tube Pulling," Vol. 18, No. 3, Sept. 1971, pp. 423427.
DISNEY, R. L. and W. E. Mitchell, "A Solution for Queues with Instantaneous Jockeying and Other Customer Selection Rules,"
Vol. 17, No. 3, Sept. 1970, pp. 315325.
DIXION, H. L., D. A. D'Esopo, and B. Lefkowitz, "A Model for Simulating an AirTransportation System," Vol. 7, No. 3, Sept.
1960, pp. 213220.
DOBBIE, J. M., "Search Theory: A Sequential Approach," Vol. 10, No. 4, Dec. 1963, pp. 323334.
DONIS, J. N., "Allocation of Resources to Randomly Occurring Opportunities," Vol. 14, No. 4, Dec. 1967, pp. 513527.
DOREY, F. D., R. D. Campbell, and R. E. Murphy, "Concept of a Logistics System," Vol. 4, No. 2, June 1957, pp. 101116.
DREBES, C, "An Approximate Solution Method for the Fixed Charge Problem," Vol. 14, No. 1, Mar. 1967, pp. 101113.
DRESCH, F., "OnLine MacroSimulation in Systems for Logistics Decision Making," Vol. 7, No. 4, Dec. 1960, pp. 447452.
DREYFUS, S. and R. Bellman, "A Bottleneck Situation Involving Interdependent Industries," Vol. 5, No. 4, Dec. 1958, pp.
307314.
DUBEY,S. D., "A New Derivation of the Logistic Distribution," Vol. 16, No. 1, Mar. 1969. pp. 37^M).
DUBEY, S. D., "On Models for Business Failure Data," Vol. 18, No. 4, Dec. 1971, pp. 561566.
DUBEY, S. D., "A Compound WeibuU Distribution," Vol. 15, No. 2, June 1968, pp. 179188.
DUBEY, S. D., "Asymptotic Efficiencies of the Moment Estimators for the Parameters of the Weibull Laws," Vol. 13, No. 3,
Sept. 1966, pp. 265288.
DUBEY, S. D., "HyperEfficient Estimator of the Location Parameter of the Weibull Laws," Vol. 13, No. 3, Sept. 1966, pp.
253264.
DUBEY, S. D., "Normal and Weibull Distributions," Vol. 14, No. 1, Mar. 1967, pp. 6979.
DUBEY, S. D., "On Some Statistical Inferences for Weibull Laws," Vol. 13, No. 3, Sept. 1966, pp. 227251.
DUBEY, S. D., "Revised Tables for Asymptotic Efficiencies of the Moment Estimators for the Parameters of the Weibull Laws,"
Vol. 14, No. 2, June 1967, pp. 261267.
DUBEY, S. D., "Some Simple Estimators for the Shape Parameter of the Weibull Laws," Vol. 14, No. 4, Dec. 1967, pp. 489512.
DUBEY, S. D., "Some Test Functions for the Parameters of the Weibull Distributions," Vol. 13, No. 2, June 1966, pp. 113128.
DUDEWICZ, E. J., "Confidence Intervals for Ranked Means," Vol. 17, No. 1, Mar. 1970, pp. 6978.
DUDEWICZ, E. J. and C. Fan, "Further Light on Nonparametric Selection Efficiency," Vol. 20, No. 4, Dec. 1973, pp. 737744.
DUTTA, S. K. and J. A. Buzacott, "Sequencing Many Jobs on a MultiPurpose Facility," Vol. 18, No. 1, Mar. 1971, pp. 7582.
DUTTA, S. K. and A. A. Cunningham, "Scheduling Jobs, with Exponentially Distributed Processing Times, on Two Machines
of a Flow Shop," Vol. 20, No. 1, Mar. 1973, pp. 6981.
DWYER, P. S., "Use of Completely Reduced Matrices in Solving Transportation Problems with Fixed Charges," Vol. 13, No. 3,
Sept. 1966, pp. 289313.
DWYER, P. S., and B. Caller, "Translating the Method of Reduced Matrices to Machines," Vol. 4, No. 1, Mar. 1957, pp. 5571.
EASTMAN, S. E., "Aircraft Loading Considerations: A Sortie Generator for Use in Planning Military Transport Operation,"
Vol. 15, No. 1, Mar. 1968, pp. 99119.
ECCLES, H. E., "A Note on Management and Logistics," Vol. 14, No. 1, Mar. 1967, p. 131.
ECCLES, H. E., "Logistics: Conditio sine qua non for NATO Defense," Vol. 8, No. 1, Mar. 1967, pp. 111116.
ECCLES, H. E., "Logistics What is it?" Vol. 1, No. 1, Mar. 1954, pp. 515.
ECCLES, H. E., "Some Command Problems and Decisions," Vol. 2, Nos. 1 & 2, Mar. June 1955, pp. 915.
ECCLES, H. E., "Some Notes on Military Theory," Vol. 15, No. 1, Mar. 1968, pp. 121122.
ECCLES, H. E., "The Logistics Aspects of Command Control Systems," Vol. 9, No. 2, June 1962, pp. 97105.
ECCLES, H. E., "The Quartermaster Corps: Organization, Supply, and Services, Volume 1: A Book Review," Vol. 1, No. 3,
Sept. 1954, pp. 207209.
ECCLES, H. E., "The Rommel Papers A Commentary," Vol. 1, No. 2, June 1954, pp. 103108.
ECCLES, H. E., "The Study of Military Management," Vol. 13, No. 4, Dec. 1966, pp. 437^145.
EGAN, J. F., L. Gleiberman, and J. Laderman, "Vessel Allocation by Linear Programming," Vol. 13, No. 3, Sept. 1966, pp.
315320.
EHRENFELD, S. and R. C. Chang, "On a Sequential Test Procedure with Delayed Observations," Vol. 19, No. 4, Dec. 1972,
pp. 651661.
EICHHORN, B. H., "Sequential Search of an Optimal Dosage: NonBayesian Methods," Vol. 20, No. 4, Dec. 1973, pp. 729736.
EISENMAN, R. L., "Alliance Games of nPersons," Vol. 13, No. 4, December 1966, pp. 403411.
EKLOF, W. D., M. Bellmore and G. L. Nemhauser, "A Decomposable Transshipment Algorithm for a Multiperiod Transpor
tation Problem," Vol. 16, No. 4, Dec. 1969, pp. 517524.
ELMAGHRABY, S. E., "A Graph Theoretic Interpretation of the Sufficiency Conditions for the ContiguousBinarySwitching
(CBS)Rule," Vol. 18, No. 3, Sept. 1971, pp. 339344.
ELMAGHRABY, S. E., "The Machine Sequencing Problem Review and Extensions," Vol. 15, No. 2, June 1968, pp. 205232.
ELMAGHRABY, S. E., "The Sequencing of 'Related' Jobs," Vol. 15, No. 1, Mar. 1968, pp. 2332.
820 CUMULATIVE INDEX FOR VOLUMES 120
ENGLISH, J. A. and E. A. Jerome, "Statistical Methods for Determining Requirements of Dental Materials," Vol. 1, No. 3,
Sept. 1954, pp. 191199.
ENSLOW, P. H., JR., "A Bibliography of Search Theory and Reconnaissance Theory Literature," Vol. 13, No. 2, June 1966,
pp. 177202.
ENZER, H., "Economic Impact and the Notion of Compensated Procurement," Vol. 15, No. 1, Mar. 1968, pp. 6379.
ENZER, H., "On Some Economic Concepts of Multiple Incentive Contracting," Vol. 15, No. 4, Dec. 1968, pp. 477489.
ENZER, H., "On Two Nonprobabilistic Utility Measures for Weapon Systems," Vol. 16, No. 1, Mar. 1969, pp. 5361.
ERCAN, S. S., "Systems Approach to the Multistage Manufacturing ConnectedUnit Situation," Vol. 19, No. 3, Sept. 1972,
pp. 493500.
ERICSON, W. A., "On the Minimization of a Certain Convex Function Arising in Applied Decision Theory," Vol. 15, No. 1,
Mar. 1968, pp. 3348.
ESELSON, L., S. Glickman, and L. Johnson, "Coding the Transportation Problem," Vol. 7, No. 2, June 1960, pp. 169183.
EVANS, G. W., II, "A Transportation and Production Model," Vol. 5, No. 2, June 1958, pp. 137154.
EVANS, G. W., II, P. Brock, and W. D. Correll, "Techniques for Evaluating Military Organization and Their Equipment,"
Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 211229.
EVANS, J. P., "On Constraint Qualifications in Nonlinear Programming," Vol. 17, No. 3, Sept. 1970, pp. 281286.
EVANS, J. P. and F. J. Gould, "Application of the GLM Technique to a Production Planning Problem," Vol. 18, No. 1, Mar.
1971, pp. 5974.
EVANS, R. V., "Inventory Control of ByProducts," Vol. 16, No. 1, Mar. 1969, pp. 8592.
EVANS, R. V., "Inventory Control of a Multiproduct System with a Limited Production Resource," Vol. 14, No. 2, June 1967,
pp. 173184.
FALKNER, C. H., "Optimal Spares for Stochastically Failing Equipment," Vol. 16, No. 3, Sept. 1969, pp. 287295.
FAN, C. and E. J. Dudewicz, "Further Light on Nonparamertic Selection Efficiency," Vol. 20, No. 4, Dec. 1973, pp. 737744.
FARRAR, D. E. and R. E. Apple, "Some Factors that Affect the Overhaul Cost of Ships: An Exercise in Statistical Cost Analysis,"
Vol. 10, No. 4, Dec. 1963, pp. 335368.
FENNELL, J. and S. Zacks, "Bayes Adaptive Control of TwoEchelon Inventory SystemsT: Development for a Special Case of
OneStation Lower Echelon and Monte Carlo Evaluation," Vol. 19, No. 1, Mar. 1972, pp. 1528.
FENNELL, J., M. Denicoff, S. E. Haber, W. H. Marlow, F. W. Segel, and H. Solomon, "The Polaris Military Essentiality Sys
tem," Vol. 11, No. 4, Dec. 1964, pp. 235257.
FENNELL, J., M. Denicoff, S. E. Haber, W. H. Marlow, and H. Solomon, "A Polaris Logistics Model," Vol. 11, No. 4, Dec.
1964, pp. 259272.
FENNELL, J. and S. Oshiro, "The Dynamics of Overhaul and Replenishment Systems for Large Equipments," Vol. 3, Nos. 1
& 2, Mar. June 1956, pp. 1943.
FENNELL, J. P., "An Automatic Addressing Device," Vol. 7, No. 4, Dec. 1960, pp. 373378.
FENNELL, J. P., M. Denicoff, and H. Solomon, "Summary of a Method for Determining the Military Worth of Spare Parts,"
Vol. 7, No. 3, Sept. 1960, pp. 221234.
FENSKE, W. J. and S. Zacks, "Sequential Determination of Inspection Epochs for Reliability Systems with General Lifetime
Distributions," Vol. 20, No. 3, Sept. 1973, pp. 377386.
FIELD, C. A., R. G Cassidy and M. J. L. Kirby, "Partial Information in Two Person Games with Random Payoffs," Vol. 20,
No. 1, Mar. 1973, pp. 4156.
FINCHIM, J. A., JR., "Expectation of Contract Incentives," Vol. 19, No. 2, June 1972, pp. 389397.
FINN, W. R. and A. H. Wilson, "Improvise or Plan?" Vol. 4, No. 4, Dec. 1957, pp. 263267.
FIRSTMAN, S. I., "An Aid to Preliminary Design of Missile Prelaunch Checkout Equipment," Vol. 9, No. 1, Mar. 1962, pp.
1729.
FIRSTMAN, S. I., "An Approximation Algorithm for an Optimum AimPoints Problem," Vol. 7, No. 2, June 1960, pp. 151167.
FISCHER, C. A. and R. Meade, Jr., "Mobile Logistics Support in the 'Passage to Freedom' Operation," Vol. 1, No. 4, Dec.
1954, pp. 258264.
FISHBURN, P. C, "Additive Utilities with Finite Sets: Applications in the Management Sciences," Vol. 14, No. 1, Mar. 1967,
pp. 113.
FISHER, I. N., "An Evaluation of Incentive Contracting Experience," Vol. 16, No. 1, Mar. 1969, pp. 63~83.
FITZPATRICK, G. R., "Programming the Procurement of Airlift and Sealift Forces: A Linear Programming Model for Analysis
of the LeastCost Mix of Strategic Deployment Systems," Vol. 14, No. 2, June 1967, pp. 241255.
FLOOD, M. M., "A Transportation Algorithm and Code," Vol. 8, No. 3, Sept. 1961, pp. 257276.
FLOOD, M. M., "Operations Research and Logistics," Vol. 5, No. 4, Dec. 1958, pp. 323335.
FLORIAN, M., P. Bratley, and P. Robillard, "Scheduling with Earliest Start and Due Date Constraints," Vol. 18, No. 4, Dec.
1971, pp. 511519.
FLORIAN, M., P. Bratley and P. Robillard, "On Sequencing with Earliest Starts and Due Dates with Application to Computing
Bounds for the (nlm/GIF max ) Problem," Vol. 20, No. 1, Mar. 1973, pp. 5767.
FOLSOM, P. L., "Military Worth and Systems Development," Vol. 7, No. 4, Dec. 1960, pp. 501511.
FORD, L. R., JR. and D. R. Fulkerson, "A PrimalDual Algorithm for the Capacitated Hitchcock Problem," Vol. 4, No. 1, Mar.
1957, pp. 4754.
CUMULATIVE INDEX FOR VOLUMES 120 821
FRANCIS, R. L. and A. V. Cabot, "Properties of a Multifacility Location Problem Involving Euclidian Distances," VoL 19,
No. 2, June 1972, pp. 335353.
FRANK, M. and P. Wolfe, "An Algorithm for Quadratic Programming," VoL 3, Nos. 1 & 2, Mar. June 1956, pp. 95110.
FRANK, P., "On Assuring Safety in Destructive Testing," Vol. 7, No. 3, Sept. 1960, pp. 257259.
FRIEDMAN, L., "Annex B: Calculation of the School Table of the Paper 'An Analysis of Stewardess Requirements and Sched
uling for a Major Domestic Airline,'" Vol. 4, No. 3, Sept. 1957, pp. 199202.
FRIEDMAN, L. and A. J. Yaspan, "Annex A: The Assignment Problem Technique" of the paper "An Analysis of Stewardess
Requirements and Scheduling for a Major Domestic Airline," Vol. 4, No. 3, Sept. 1957, pp. 193197.
FRISCH, H., "Consumption, the Rate of Interest and the Rate of Growth in the von Neumann Model," Vol. 16, No. 4, Dec.
1969, pp. 459484.
FUKUDA, Y., "Optimal Disposal Policies," Vol. 8, No. 3, Sept. 1961, pp. 221227.
FULKERSON, D. R. and G. B. Dantzig, "Computation of Maximal Flows in Networks," Vol. 2, No. 4, Dec. 1955, pp. 277283.
FULKERSON, D. R. and G. B. Dantzig, "Minimizing the Number of Tankers to Meet a Fixed Schedule," Vol. 1, No. 3, Sept.
1954, pp. 217222.
FULKERSON, D. R. and L. R. Ford, Jr., "A PrimalDual Algorithm for the Capacitated Hitchcock Problem," Vol. 4, No. 1, Mar.
1957, pp. 4754.
GADDUM, J. W., A. J. Hoffman, and D. Sokolowsky, "On the Solution of the Caterer Problem," Vol. 1, No. 3, Sept. 1954, pp.
223229.
GAFARIAN, A. V. and C. J. Ancker, Jr., "Queing with Reneging and Multiple Heterogeneous Servers," Vol. 10, No. 2, June
1963, pp. 125149.
"GAFARIAN, A. V. and C. J. Ancker, Jr., "The Distribution of Rounds Fired in Stochastic Duels," Vol. 11, No. 4, Dec. 1964,
pp. 303327.
GAFARIAN, A. V. and C. J. Ancker, Jr., "The Distribution of the TimeDuration of Stochastic Duels," Vol. 12, Nos. 3 & 4,
Sept.Dec. 1965, pp. 275294.
GAINEN, L., E. D. Stanley, and D. P. Honig, "Linear Programming in Bid Evaluation," VoL 1, No. 1, Mar. 1954, pp. 4954.
GALE, D., "A Note on Global Instability of Competitive Equilibrium," Vol. 10, No. 1, Mar. 1963, pp. 8187.
GALE, D., "The Basic Theorems of Real Linear Equations, Inequalities, Linear Programming, and Game Theory," VoL 3,
No. 3, Sept. 1956, pp. 193200.
GALLER, B. and P. S. Dwyer, "Translating the Method of Reduced Matrices to Machines," VoL 4, No. 1, Mar. 1957, pp. 5571.
GARFINKEL, R. S. and M. R. Rao, "The Bottleneck Transportation Problem," VoL 18, No. 4, Dec. 1971, pp. 465472.
GARFUNKEL, I. M. and J. E. Walsh, "Method for FirstStage Evaluation of Complex ManMachine Systems," Vol. 7, No. 1,
Mar. 1960, pp. 1319.
GARG, R. C, "Analytical Study of a Complex System Having Two Types of Components," Vol. 10, No. 3, Sept. 1963, pp. 263269.
GARG, R. C. and C. Mohan, "Decision on Retention of Excess Stock Following a Normal Probability Law of Obsolescence
and Deterioration," VoL 8, No. 3, Sept. 1961, pp. 229234.
GARRETT, J. H., JR., "Characteristics of Usage of Supply Items Aboard Naval Ships and the Significance to Supply Manage
ment," VoL 5, No. 4, Dec. 1958, pp. 287306.
GASCHUTZ, G. K., "Suboptimal Algorithms for the Quadratic Assignment Problem," Vol. 15, No. 1, Mar. 1968, pp. 4962.
GASS, S. I., "On the Distribution of Manhours to Meet Scheduled Requirements," Vol. 4, No. 1, Mar. 1957, pp. 1725.
GASS, S. and T. Saaty, "The Computational Algorithm for the Parametric Objective Function," Vol. 2, Nos. 1 & 2, Mar. June
1955, pp. 3945.
GASSNER, B. J., "Cycling in the Transportation Problem," VoL 11, No. 1, Mar. 1964, pp. 4358.
GAVER, D. P., J. M. Burt, Jr., and M. Perlas, "Simple Stochastic Networks: Some Problems and Procedures," Vol. 17, No. 4,
Dec. 1970, pp. 439459.
GAVER, D. P. and G. S. Shedler, "Control Variable Methods in the Simulation of a Model of a Multiprogrammed Computer Sys
tem," VoL 18, No. 4, Dec. 1971, pp. 435450.
GAVER, D. P., JR., "A Comparison of Queue Disciplines When Service Orientation Times Occur," VoL 10, No. 3, Sept. 1963,
pp. 219235.
GAVER, D. P., JR., "RenewalTheoretic Analysis of a TwoBin Inventory Control Policy," Vol. 6, No. 2, June 1959, pp. 141163.
GAVER, D. P., JR., "Statistical Estimation in a Problem of System Reliability," Vol. 14, No. 4, Dec. 1967, pp. 473488.
GEBHARD, R. F., "A Queueing Process with Bilevel Hysteretic ServiceRate Control," Vol. 14, No. 1, Mar. 1967, pp. 5567.
GEISLER, M. A., "A First Experiment in Logistics System Simulation," Vol. 7, No. 1, Mar. 1960, pp. 2144.
GEISLER, M. A., "Relationships Between Weapons and Logistics Expenditures," Vol. 4, No. 4, Dec. 1957, pp. 335346.
GEISLER, M. A., "Some Principles for a DataProcessing System in Logistics," Vol. 5, No. 2, June 1958, pp. 95105.
GEISLER, M. A., "Statistical Properties of Selected Inventory Models," Vol. 9, No. 2, June 1962, pp. 137156.
GEISLER, M. A., "The Use of ManMachine Simulation for Support Planning," VoL 7, No. 4, Dec. 1960, pp. 421428.
GEISLER, M. A., W. W. Haythorn, and W. A. Steger, "Simulation and the Logistics Systems Laboratory," Vol. 10, No. 1, Mar.
1963, pp. 2354.
GEISLER, M. A., and J. W. Petersen, "The Costs of Alternative Air Base Stocking and Requisitioning Policies," VoL 2, Nos.
1 & 2, Mar. June 1955, pp. 6981.
GELKE, J. J., A. S. Rhode and F. X. Cook, "Impact of an All Volunteer Force upon the Navy in the 19721973 Timeframe,"
VoL 19, No. 1, Mar. 1972, pp. 4375.
822 CUMULATIVE INDEX FOR VOLUMES 120
GEORGE, L. L. and A. C. Agrawal, "Estimation of a Hidden Service Distribution of an M/G/oo System," Vol. 20, No. 3, Sept. 1973,
pp. 549555.
GERSON, G. and R. G. Brown, "Decision Rules for Equal Shortage Policies," Vol. 17, No. 3, Sept. 1970, pp. 351358.
GHARE, P. M., D. C. Montgomery and W. C. Turner, "Optimal Interdiction Policy for a Flow Network," Vol. 18, No. 1, Mar.
1971, pp. 3745.
GHARE, P. M. and W. J. Kennedy, Jr., "Optimum Adjustment Policy for a Product with Two Quality Characteristics," Vol. 20,
No. 4, Dec. 1973, pp. 785791.
GIFFLER, B., "Determining and Optimum Reject Allowance," Vol. 7, No. 2, June 1960, pp. 201206.
GIFFLER, B., "Schedule Algebra: A Progress Report," Vol. 15, No. 2, June 1968, pp. 255280.
GIFFLER, B, "Scheduling General Production Systems Using Schedule Algebra," Vol. 10, No. 3, Sept. 1963, pp. 237255.
GILBERT, J. C, "A Method of Resource Allocation Using Demand Preference," Vol. 11, Nos. 2 and 3, JuneSept. 1964, pp.
217225.
GILLILAND, D. C, "On Maximization of the Integral of a BellShaped Function Over a Symmetric Set," Vol. 15, No. 4, Dec.
1968, pp. 507515.
CLEAVES, V. B., "Cyclic Scheduling and Combinatorial Topology; Assignment and Routing of Motive Power to Meet Scheduling
and Maintenance Requirements," Part I, "A Statement of the Operating Problem of the Frisco Railroad," Vol. 4, No. 3,
Sept. 1957, pp. 203205.
GLEIBERMAN, L., F. F. Egan, and J. Laderman, "Vessel Allocation by Linear Programming," Vol. 13, No. 3, September 1966,
pp. 315320.
GLICKSMAN, S., L. Johnson, and L. Eselson, "Coding the Transportation Problem," Vol. 7, No. 2, June 1960, pp. 169183.
GLOVER, F., A. Charnes and D. Klingman, "The Lower Bounded and Partial Upper Bounded Distribution Model," Vol. 18,
No. 2, June 1971, pp. 277281.
GLOVER, F., "Maximum Matching in a Convex Bipartite Graph," Vol. 14, No. 3, Sept. 1967, pp. 313316.
GLUSS, B., "An Alternative Solution to the 'Lost at Sea' Problem," Vol. 8, No. 1, Mar. 1961, pp. 117121.
GLUSS, B., "An Optimal Inventory Solution for Some Specific Demand Distribution," Vol. 7, No. 1, Mar. 1960, pp. 4548.
GLUSS, B., "Approximately Optimal OneDimensional Search Policies in Which Search Costs Vary Through Time," Vol. 8,
No. 3, Sept. 1961, pp. 277283.
GLUSS, B., "The Minimax Path in a Search for a Circle in a Plane," Vol. 8, No. 4, Dec. 1961, pp. 357360.
(iOLDFARB, D. and S. Zacks, "Survival Probabilities in Crossing a Field Containing Absorption Points," Vol. 13, No. 1, Mar.
1966, pp. 3548.
GOLDMAN, A. S., "Problems in Life Cycle Support Cost Estimation," Vol. 16, No. l,Mar. 1969, pp. 111120.
GOMORY, R. E. and A. J. Hoffman, "On the Convergence of an IntegerProgramming Process," Vol. 10, No. 2, June 1963,
pp. 121123.
GOODMAN, A. F., "Extended Iterative Weighted Least Squares: Estimation of a Linear Model in the Presence of Complica
tions," Vol. 18, No. 2, June 1971, pp. 243276.
GOODMAN, A. F., "The Interface of Computer Science and Statistics," Vol. 18, No. 2, June 1971 , pp. 215229.
GOODMAN, I. F., "Statistical Quality Control of Information," Vol. 17, No. 3, Sept. 1970, pp. 389396.
GOULD, F. J. and J. P. Evans, "Application of the GLM Technique to a Production Planning Problem," Vol. 18, No. 1, Mar. 1971,
pp. 5974.
GOURARY, M. H., "A Simple Rule for the Consolidation of Allowance Lists," Vol. 5, No. 1, Mar. 1958, pp. 115.
GOURARY, M. H., "An Optimum Allowance List Model," Vol. 3, No. 3, Sept. 1956, pp. 177192.
GOURARY, M. H., R. Lewis, and F. Neeland, "An Inventory Control Bibliography," Vol. 3, No. 4, Dec. 1956, pp. 295303.
GOYAL, J. K. and S. K. Gupta, "Queues with Batch Poisson Arrivals and HyperExponential Service," Vol. 12, Nos. 3 & 4,
Sept.Dec. 1965, pp. 323329.
GRAVES, G. W., "A Complete Constructive Algorithm for the General Mixed Linear Programming Problem," Vol. 12, No. 1,
Mar. 1965, pp. 134.
GREENBERG, H., "A Note on a Modified PrimalDual Algorithm to Speed Convergence in Solving Linear Programs," Vol. 16,
No. 2, June 1969, pp. 271273.
GREENBERG, H., "A Quadratic Assignment Problem Without Column Constraints," Vol. 16, No. 3, Sept. 1969, pp. 417421.
GREENBERG, H., "Stock Level Distributions for (s, S) Inventory Problems," Vol. 11, No. 4, Dec. 1964, pp. 343349.
GREENWOOD, J. A., "Issue Priority: Last In First Out (LIFO) vs. First In First Out (FIFO) as a Method of Issuing Items from
Supply Storage," Vol. 2, No. 4, Dec. 1955, pp. 251268.
GREISMER, J. H. "Toward a Study of Bidding Processes: Part IV Games with Unknown Costs," Vol. 14, No. 4, Dec. 1967,
pp. 415433.
GREISMER, J. H. and M. Shubik, "Toward a Study of Bidding Processes: Some ConstantSum Games," Vol. 10, No. 1, Mar. 1963,
pp. 1121.
GREISMER, J.H. and M. Shubik, "Toward a Study of Bidding Processes, Part II: Games with Capacity Limitations," Vol.
10, No. 2, June 1963, pp. 151173.
GREISMER, J. H. and M. Shubik, "Toward a Study of Bidding Processes, Part III: Some Special Models," Vol. 10, No. 3, Sept.
1963, pp. 199217.
GRINOLD, R. C, "The Payment Scheduling Problem," Vol. 19, No. 1, Mar. 1972, pp. 123136.
CUMULATIVE INDEX FOR VOLUMES 120 823
GROSS, D. and R. M. Soland, "A Branch and Bound Algorithm for Allocation Problems in Which Constraint Coefficients De
pend upon Decision Variables," Vol. 16, No. 2, June 1969, pp. 157174.
GROSS, D. and A. Soriano, "On the Economic Application of Airlift to Product Distribution and Its Impact on Inventory Levels,"
Vol. 19, No. 3, Sept. 1972, pp. 501507.
GRUNSPAN, M. and M. E. Thomas, "Hyperbolic Integer Programming," Vol. 20, No. 2, June 1973, pp. 341356.
GSELLMAN, L., G. Bennington and S. Lubore, "An Economic Model for Planning Strategic Mobility Posture," Vol. 19, No. 3,
Sept. 1972, pp. 461470.
GUENTHER, W. C, "Tolerance Intervals for Univariate Distributions," Vol. 19, No. 2, June 1972. pp. 309333.
GUENTHER, W. C. "On the Use of Standard Tables to Obtain DodgeRomig LTPD Sampling Inspection Plans," Vol. 18, No. 4,
Dec. 1971, pp. 531542.
GUPTA, S. K. and J. K. Goyal, "Queues with Batch Poisson Arrivals and HyperExponential Service," Vol. 12, Nos. 3 & 4, Sept.
Dec. 1965, pp. 323329.
GUSTAFSON, S. A. and K. O. Kortanek, "Numerical Treatment of a Class of SemiInfinite Programming," Vol. 20, No. 3, Sept.
1973. pp. 477504.
GUTHRIE, D., JR., "Relationships Among Potential Sorties, Ground Support, and Aircraft Reliability." Vol. 15, No. 4, Dec.
1968, pp. 491506.
GUTHRIE, D., JR. and F. R. Collins, Jr., "A Model for the Analysis of AEW and CAP Aircraft Availability." Vol. 10. No. 1,
Mar. 1963, pp. 7379.
HABER, S., R. Sitgreaves and H. Solomon, "A Demand Prediction Technique for Items in Military Inventory Systems," Vol.
16, No. 3, Sept. 1969, pp. 297308.
HABER, S. E., "Simulation of MultiEchelon MacroInventory Policies," Vol. 18, No. 1, Mar. 1971, pp. 119134.
HABER, S. E. and R. Sitgreaves. "A Methodology for Estimating Expected Usage of Repair Parts with Application to Parts
with No Usage History," Vol. 17. No. 4. Dec. 1970. pp. 535546.
HABER, S. E. and R. Sitgreaves, "A Unified Model for Demand Prediction in the Context of Provisioning and Replenishment,"
Vol. 19, No. 1, Mar. 1972, pp. 2942.
HABER, S. E., F. W. Segel and H. Solomon, "Statistical Auditing of LargeScale Management Information Systems," Vol.
19, No. 3, Sep. 1972, pp. 449459.
HABER, S. E., "A Comparison of Usage Data Among Types of Aircraft," Vol. 14, No. 3, Sept. 1967, pp. 399410.
HABER, S. E., M. Denicoff, J. Fennell, W. H. Marlow, F. W. Segel and H. Solomon, "The Polaris Military Essentiality Sys
tem," Vol. 11, No. 4, Dec. 1964, pp. 235257.
HABER, S. E.. M. Denicoff, J. Fennell, W. H. Marlow and H. Solomon. "A Polaris Logistics Model," Vol. 11, No. 4, Dec. 1964,
pp. 259272.
HADLEY, G. F. and M. A. Simonnard, "A Simplified TwoPhase Technique for the Simplex Method," Vol. 6, No. 3, Sept. 1959,
pp. 221226.
HADLEY, G. F. and M. A. Simmonard, "Maximum Number of Iterations in the Transportation Problem," Vol. 6. No. 2, June
1959, pp. 125129.
HADLEY, G. and T. M. Whitin, "A Model for Procurement. Allocation, and Redistribution for Low Demand Items," Vol. 8,
No. 4, Dec. 1961, pp. 395414.
HADLEY, G. and T. M. Whitin, "Budget Constraints in Logistics Models," Vol. 8, No. 3, Sept. 1961. pp. 215220.
HALE, J. K. and R. Reed, "A Formulation of the Decision Problem for a Class of Systems," Vol. 3, No. 4, Dec. 1956, pp. 259277.
HALE, J. K. and H. H. Wicke, "An Application of Game Theory to Special Weapons Evaluation," Vol. 4, No. 4, Dec. 1957, pp.
347356.
HALL, W., M. Paulsen and H. Bremmer, "Experiences With the Bid Evaluation Problem (Abstract)," Vol. 4, No. 1, Mar. 1957,
p. 27.
HAMILTON, J. E., "Realism in Empirical Logistics Research," Vol. 7. No. 4, Dec. 1960, pp. 493499.
HAMMER, P. L., "Communication on 'The Bottleneck Transportation Problem' and 'Some Remarks on the Time Transporta
tion Problem'," Vol. 18, No. 4, Dec. 1971, pp. 487490.
HAMMER, P. L., "Time Minimizing Transportation Problems," Vol. 16, No. 3, Sept. 1969, pp. 345357.
HANDLER, G., "Termination Policies for a TwoState Stochastic Process," Vol. 19, No. 2, Jun. 1972, pp. 281292.
HANSSMAN, F. and J. F. McCloskey, "An Analysis of Stewardess Requirements and Scheduling for a Major Domestic Air
line," Vol. 4, No. 3, Sept. 1957, pp. 183192.
HARARY, F., "A Matrix Criterion for Structural Balance," Vol. 7, No. 2, June 1970, pp. 195199.
HARARY, F. and M. Richardson, "A Matrix Algorithm for Solutions and rBases of a Finite Irreflexive Relation," Vol. 6, No.
4, Dec. 1959, pp. 307314.
HARRIS, B., "The Probability of Survival of a Subterranean Target Under Intensive Attack," Vol. 14, No. 4, Dec. 1967, pp.
435451.
HARRIS, C. M., "On Queues with StateDependent Erlang Service," Vol. 18, No. 1, Mar. 1971, pp. 103110.
HARRIS, C. M. and N. D. Singpurwalla, "On Estimation in Weibull Distributions with Random Scale Parameters," Vol. 16,
No. 3, Sept. 1969, pp. 405410.
HARRIS, C. M., "A Queueing System with Multiple Service Time Distributions," Vol. 14, No. 2, June 1967. pp. 231239.
824 CUMULATIVE INDEX FOR VOLUMES 120
HARRIS, C. M., "Queues with Stochastic Service Rates," Vol. 14, No. 2, June 1967, pp. 219230.
HARRIS, M. Y., "A Mutual PrimalDual Linear Programming Algorithm," Vol. 17, No. 2, June 1970, pp. 199206.
HARRISON, S. and J. E. Jacoby, "MultiVariable Experimentation and Simulation Models," Vol. 9, No. 2, June 1962, pp. 121
136.
HARTMAN, J. K., "Some Experiments in Global Optimization," Vol. 20, No. 3, Sep. 1973, pp. 569576.
HARTMAN, J. K. and L. S. Lasdon, "A Generalized Upper Bounding Method for Doubly Coupled Linear Programs," Vol. 17,
No. 4, Dec. 1970, pp. 411^29.
HAYES, J. D., "Logistics The Word," Vol. 1, No. 3, Sept. 1954, pp. 200202.
HAYTHORN, W. W., M. A. Geisler, and W. A. Steger, "Simulation and the Logistics System Laboratory," Vol. 10, No. 1, Mar.
1963, pp. 2354.
HECK, H. and S. Roberts, "A Note on the Extension of a Result on Scheduling with Secondary Criteria," Vol. 19, No. 2, Jun.
1972, pp. 403405.
HEDETNIEMI, S., "On Minimum Walks in Graphs," Vol. 15, No. 3, Sept. 1968, pp. 453458.
HEIDER, C. H., "An NStep, 2 Variable Search Algorithm for the Component Placement Problem," Vol. 20, No. 4, Dec. 1973,
pp. 699724.
HEIMANN, D. and M. F. Neuts, "The Single Server Queue in Discrete TimeNumerical Analysis IV," Vol. 20, No. 4, Dec. 1973,
pp. 753766.
HELLER, I., "Constraint Matrices of Transportation — Type Problems (Abstract)," Vol. 4, No. 1, Mar. 1957, pp. 7376.
HEMPLEY, R. B. and R. A. Agnew, "Finite Statistical Games and Linear Programming," Vol. 18, No. 1, Mar. 1971, pp. 99102.
HENIN, C, "Computational Techniques for Optimizing Systems with Standby Redundancy," Vol. 19, No. 2, June 1972, pp.
293308.
HENIN, C. G., "Optimal Allocation of Unreliable Components for Maximizing Expected Profit Over Time," Vol. 20, No. 3,
Sep. 1973, pp. 395403.
HENN, C. L., "Multinational Logistics in the Nuclear Age," Vol. 4, No. 2, June 1957, pp. 117130.
HERRON, D. P., "Use of Dimensionless Ratios to Determine MinimumCost Inventory Quantities," Vol. 13, No. 2, June 1966,
pp. 167176.
HERSHKOWITZ, M., "A Computational Note on Von Newmann's Algorithm for Determining Optimal Strategy," Vol. 2, No. 1,
Mar. 1964, pp. 75 78.
HERSHKOWITZ, M. and S. B. Noble, "Finding the Inverse and Connections of a Type of Large Sparse Matrix," Vol. 12, No.
l,Mar. 1965, pp. 119132.
HESELDEN, G. P. M. and E. M. L. Beale, "An Approximate Method of Solving Blotto Games," Vol. 9, No. 2, June 1962, pp.
6579.
HETTER, F. L., "Navy Stratification and Fractionation for Improvement of Inventory Management," Vol. 1, No. 2, June 1954,
pp. 7578.
HIGGINS, J. W. and R. S. Weinberg, "The Feasibility of a Global Airlift," Vol. 6, No. 2, June 1959, pp. 89110.
HILDRETH, C, "A Quadratic Programming Procedure," Vol. 4, No. 1, Mar. 1957, pp. 7985.
HIRSCH, W. M., "Cannibalization in Multicomponent Systems and the Theory of Reliability," Vol. 15, No. 3, Sept. 1968, pp.
331360.
HIRSCH, W. M., "The Fixed Charge Problem," Vol. 15, No. 3, Sept. 1968, pp. 413424.
HITCHCOCK, D. F. and J. B. MacQueen, "On Computing the Expected Discounted Return in a Markov Chain," Vol. 17, No.
2, June 1970, pp. 237241.
HO, C, "A Note on the Calculation of Expected TimeWeighted Backorders Over A Given Interval," Vol. 17, No. 4, Dec. 1970,
pp. 555559.
HOCHBERG, M., "Generalized Multicomponent Systems under Cannibalization," Vol. 20, No. 4, Dec. 1973, pp. 585605.
HOFFMAN, A. J., "On Abstract Dual Linear Programs," Vol. 10, No. 4, Dec. 1963, pp. 369373.
HOFFMAN, A. J., J. W. Gaddum, and D. Sokolowsky, "On the Solution of the Caterer Problem," Vol. 1, No. 3, Sept. 1954,
pp. 223229.
HOFFMAN, A. J. and R. E. Gomory, "On the Convergence of an IntegerProgramming Process," Vol. 10, No. 2, June 1963,
pp. 121123.
HOFFMAN, A. J. and H. M. Markowitz, "Shortest Path, Assignment, and Transportation Problems," Vol. 10, No. 4, Dec. 1963,
pp. 375379.
HOLLADAY, J. C, "Aircraft Loading Considerations: A Sortie Generator for Use in Planning Military Transport Operations,"
Vol. 15, No. 1, Mar. 1968, pp. 99119.
HOLLADAY, J., "Some Transportation Problems and Techniques for Solving Them," Vol. 11, No. 1, Mar. 1964, pp. 1542.
HOLLINGSHEAD, E. F., "The Application of Statistical Techniques to Management of Overseas Supply Operations," Vol. 1,
No. 2, June 1954, pp. 8289.
HOLLOWAY, C. A., "A Mathematical Programming Approach to Identification and Optimization of a Class of Unknown Sys
tems," Vol. 19, No. 4, Dec. 1972, pp. 663679.
HONIG, D. P., L. Gainen, and E. D. Stanley, "Linear Programming in Bid Evaluations," Vol. 1, No. 1, Mar. 1954, pp. 4954.
HORNE, R. C, "Developing an Engineered Productivity Standard," Vol. 1, No. 3, Sept. 1954, pp. 203206.
CUMULATIVE INDEX FOR VOLUMES 120 g25
HOTTENSTEIN, M. P. and J. E. Day, "Review of Sequencing Research," VoL 17, No. 1, Mar. 1970, pp. 1139.
HOUSTON, B. F. and R. A. Huffman, "A Technique which Combines Modified Pattern Search Methods with Composite Designs
and Polynomial Constraints to Solve Constrained Optimization Problems," Vol. 18, No. 1, Mar. 1971, pp. 9198.
HOWARD, G. T., "Optimal Capacity Expansion," Vol. 15, No. 4, Dec. 1968, pp. 535550.
HOWES, D. R. and R. M. Thrall, "A Theory of Ideal Linear Weights for Heterogeneous Combat Forces," Vol. 20, No. 4, Dec.
1973, pp. 645659.
HU, T. C, "MinimumCost Flows in ConvexCost Networks," Vol. 13, No. 1 March 1966, pp. 19.
HU,T. C. and W. Prager, "Network Analysis of Production Smoothing," Vol. 6, No. 1, Mar. 1959, pp. 1723.
HUFFMAN, R. A. and B. F. Houston, "A Technique which Combines Modified Pattern Search Methods with Composite Designs
and Polynomial Constraints to Solve Constrained Optimization Problems," Vol. 18, No. 1, Mar. 1971, pp. 9198.
HUNT, R. B. and E. F. Rosholdt, "Determining Merchant Shipping Requirements in Integrated Military Planning," Vol. 7,
No. 4, Dec. 1960, pp. 545575.
HUNT, R. B. and E. F. Rosholdt, "The Concepts of Notional Ship and Notional Value in Logistics Capability Studies Involving
Merchant Ships," Vol. 7, No. 1, Mar. 1960, pp. 16.
HUNTER, L. and F. Proschan, "Replacement When Constant Failure Rate Precedes Wearout," Vol. 8, No. 2, June 1961, pp.
127136.
HURWICZ, L., K. J. Arrow and H. Uzawa, "Constraint Qualifications in Maximization Problems," Vol. 8, No. 2, June 1961, pp.
175191.
IGLEHART, D. L. and S. C. Jaquette, "MultiClass Inventory Models with Demand a Function of Inventory Level," Vol. 16,
No. 4, Dec. 1969, pp. 495502.
IGLEHART, D. L. and R. C. Morey, "Optimal Policies for MultiEchelon Inventory System with Demand Forecasts," Vol. 18.
with Unknown Dependent Demands," Vol. 16, No. 4, Dec. 1969, pp. 485493.
IGLEHART, D. L. and R. C. Morey, "Optimal Polities for MultiEchelon Inventory System with Demand Forecasts," Vol. 18,
No. l,Mar. 1971, pp. 115118.
UIRI, Y. and G. L. Thompson, "Mathematical Control Theory Solution of an Interactive Accounting Flows Model," Vol. 19,
No. 3, Sept. 1972, pp. 411422.
ISAACS, R., "The Problem of Aiming and Evasion," Vol. 2, Nos. 1 & 2, Mar. June 1955, pp. 4767.
ISBELL, J. R., "An Optimal Search Pattern," Vol. 4, No. 4, Dec. 1957, pp. 357359.
ISBELL, J. R., "Pursuit Around a Hole," Vol. 14, No. 4, Dec. 1967, pp. 569571.
ISBELL, J. R. and W. H. Marlow, "Attrition Games," Vol. 3, Nos. 1 & 2, Mar. June 1956, pp. 7194.
JACKSON, J. R., "An Extension of Johnson's Results on Job Lot Scheduling," Vol. 3, No. 3, Sept. 1956, pp. 201204.
JACKSON, J. R., "Multiple Servers with Limited Waiting Space," Vol. 5, No. 4, Dec. 1958, pp. 315321.
JACKSON, J. R., "Simulation Research on Job Shop Production," Vol. 4, No. 4, Dec. 1957, pp. 287295.
JACKSON, J. R., "Some Problems in Queueing with Dynamic Priorities," Vol. 7, No. 3, Sept. 1960, pp. 235249.
JACKSON, J. R., "Some Rectangular Games with a TwoMan Team," Vol. 14, No. 1, Mar. 1967, pp. 2341.
JACKSON, J. R., "WaitingTime Distributions for Queues with Dynamic Priorities," Vol. 9, No. 1, Mar. 1962, pp. 3136.
JACKSON, J. T. Ross, "The ManyProduct Cargo Loading Problem," Vol. 14, No. 3, Sept. 1967, pp. 381390.
JACOBS, W., "Loss of Accuracy in Simplex Computations," Vol. 4, No. 1, Mar. 1957, pp. 8994.
JACOBS, W., "The Caterer Problem," Vol. 1, No. 2, June 1954, pp. 154165.
JACOBY, J. E. and S. Harrison, "MultiVariable Experimentation and Simulation Models," Vol. 9, No. 2, June 1962, pp. 121136.
JAGANNATHAN, R., W. Dent and M. R. Rao, "Parametric Linear Programming: Some Special Cases," Vol. 20, No. 4, Dec. 1973,
pp. 725728.
JAIN, H. C, "Queuing Problem with Limited Waiting Space," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 245252.
JAMES, P. M., "Derivation and Application of Unit Cost Expressions Perturbed by Design Changes," Vol. 15, No. 3, Sept. 1968,
pp. 459468.
JAMES, P. M. "Analysis of the Turbulent Regime of the Progress Curve When New Learning Additions Have Variable Slopes,"
Vol. 15, No. 3, Sept. 1968, pp. 595604.
JAQUETTE, S. C, "Suboptimal Ordering Policies Under the Full Cost Criterion," Vol. 17, No. 1, Mar. 1970, pp. 131132.
JAQUETTE, S. C. and R. W. Burton, "The Initial Provisioning Decision for Insurance Type Items," Vol. 20, No. 1, Mar. 1973,
pp. 123146.
JAQUETTE, S. C. and D. L. Iglehart, "MultiClass Inventory Models with Demand a Function of Inventory Level," Vol. 16,
No. 4, Dec. 1969, pp. 495^502.
JAQUETTE, S. C. and D. L. Iglehart, "Optimal Policies Under the Shortage Probability Criterion for an Inventory Model with
Unknown Dependent Demands," Vol. 16, No. 4, Dec. 1969, pp. 485493.
JARVIS, J. J., "On the Equivalence between the NodeArc and ArcChain Formulations for the MultiCommodity Maximal
Flow Problem," Vol. 16, No. 4, Dec. 1969, pp. 525529.
JARVIS, J. J. and J. B. Tindall, "Minimal Disconnecting Sets in Directed MultiCommodity Networks," Vol. 19, No. 4, Dec. 1972,
pp. 681690.
JAYACHANDRAN, T., W. E. Coleman and D. R. Barr, "A Note on a Comparison of Confidence Interval Techniques in Trun
cated Life Tests," Vol. 18, No. 4, Dec. 1971, pp. 567^569.
826 CUMULATIVE INDEX FOR VOLUMES 120
JEROME, E. A. and J. A. English, "Statistical Methods for Determining Requirements of Dental Materials," Vol. 1, No. 3, Sept.
1954, pp. 191199.
JEWELL, W. S., "Warehousing and Distribution of a Seasonal Product," Vol. 4, No. 1, Mar. 1957, pp. 2934.
JOHNSON, J. W.. "On Stock Selection at Spare Parts Stores Sections," Vol. 9, No. 1, Mar. 1962, pp. 4959.
JOHNSON, L., S. Glicksman and L. Eselson, "Coding the Transportation Problem," Vol. 7, No. 2, June 1960, pp. 169183.
JOHNSON, S. M., "Optimal Two and Three Stage Production Schedules with Setup Times Included," Vol. 1, No. 1, Mar. 1954,
pp. 6168.
JOKSCH, H. C. "Programming with Fractional Linear Objective Functions," Vol. 11, Nos. 2 and 3, JuneSept. 1964, pp. 197204.
JUNCOSA. M. L. and R. E. Kalaba, "CommunicationTransportation Networks (Abstract)," Vol. 4, No. 3, Sept. 1957, pp. 221
222.
JUNCOSA. M. L., "Statistical Concepts in Computational Mathematics," Vol. 18, No. 2, Jun. 1971, pp. 231242.
KALABA, R. E. and M. L. Juncosa, "CommunicationTransportation Networks (Abstract)," Vol. 4, No. 3, Sept. 1957, pp. 221
222.
KALMAN, P. I., "A Stochastic Constrained Optimal Replacement Model," Vol. 17, No. 4, Dec. 1970, pp. 547553.
KALYMON, B. A., "Structured Markovian Decision Problems," Vol. 20, No. 1, Mar. 1973, pp. 111.
KAPLAN, A. J., "The Relationship Between Decision Variables and Penalty Cost Parameter in (Q, R) Inventory Models,"
Vol. 17, No. 2, June 1970, pp. 253258.
KAPLAN, A. J., "A Stock Redistribution Model," Vol. 20, No. 2, June 1973, pp. 231239.
KAPLAN, S., "Readiness and the Optimal Redeployment of Resources," Vol. 20, No. 4, Dec. 1973, pp. 625638.
KAPUR, K. C, "On MaxMin Problems," Vol. 20, No. 4, Dec. 1973, pp. 639644.
KARLIN, S., "The Structure of Dynamic Programming Models," Vol. 2, No. 4, Dec. 1955, pp. 285 294.
KARLIN, S., W. E. Pruitt, and W. G. Madow, "On Choosing Combinations of Weapons," Vol. 10, No. 2, June 1963, pp. 95
119.
KARR. H. W., "A Method of Estimating SparePart Essentiality," Vol. 5, No. 1, Mar. 1958, pp. 2942.
KARR, H. W., W. W. Fain, and J. B. Fain, "A Tactical Warfare Simulation Program" Vol. 13, No. 4, December 1966, pp. 413
436.
KARREMAN, H. F., "Programming the Supply of a Stratigjc Material Part I: A Nonstochastic Model," Vol. 7, No. 3, Sept.
1960, pp. 261279.
KARREMAN, H., "The Probability of Survival of a Subterranean Target Under Intensive Attack," Vol. 14, No. 4, Dec. 1967,
pp. 435451.
KARUSH, W., "A Theorem in Convex Programming." Vol. 6, No. 3, Sept. 1959, pp. 245260.
KARUSH, W. and A. Vazsonyl, "Mathematical Programming and Employment Scheduling," Vol. 4, No. 4, Dec. 1957, pp. 297
320.
KEENEY, R. L., "QuasiSeparable Utility Functions," Vol. 15, No. 4, Dec. 1968, pp. 551565.
KELLEY, J. E., JR., "A Dynamic Transportation Model," Vol. 2, No. 3, Sept. 1955, pp. 175180.
KELLEY, J. E., JR., "A Threshold Method for Linear Programming," Vol. 4, No. 1, Mar. 1957, pp. 3545.
KENDRICK, J., "Changing OutputInput Relations within the National Economic Accounts," Vol. 7, No. 4, Dec. 1960, pp. 393
400.
KENNEDY, W. J., Jr. and P. M. Ghare, "Optimum Adjustment Policy for a Product with Two Quality Characteristics," Vol.
20, No. 4, Dec. 1973, pp. 785791.
KESWANI, A. K., M. S. Bazaraa and D. C. Montgomery, "Inventory Models with a Mixture of Backorders and Lost Sales,"
Vol. 20, No. 2, Jun. 1973, pp. 255263.
KHUMAWALA, B. M., "An Efficient Heuristic Procedure for the Uncapacitated Warehouse Location Problem," Vol. 20, No. 1,
Mar. 1973, pp. 109121.
KIEFER, J. and J. Wolfowitz, "Sequential Tests of Hypotheses About the Mean Occurrence Time of a Continuous Parameter
Poisson Process," Vol. 3, No. 3, Sept. 1956, pp. 205220.
KIRBY, M. J. L., C. A. Field and R. G. Cassidy, "Partial Information in Two Person Games with Random Payoffs," Vol. 20, No. 1,
Mar. 1973, pp. 4156.
KLEIN, M., "A Note on Sequential Search," Vol. 15, No. 3, Sept. 1968, pp. 469475.
KLEIN, M., "Some Production Planning Problems." Vol. 4, No. 4, Dec. 1957, pp. 269286.
KLEIN, M. and L. Rosenberg, "Deterioration of Inventory and Equipment," Vol. 7, No. 1, Mar. 1960, pp. 4962.
KLEIN, M. and C. Derman, "Surveillance of MultiComponent Systems: A Stochastic Traveling Salesman's Problem," Vol.
13, No. 2, June 1966, pp. 103111.
KLEINROCK, L., "A Conservation Law for a Wide Class of Queueing Disciplines," Vol. 12, No. 2, June 1965; pp. 181192.
KLEINROCK, L., "A Delay Dependent Queue Discipline," Vol. 11, No. 4, Dec. 1964, pp. 329341.
KLEINROCK, L., "Analysis of TimeShared Processor," Vol. 11, No. 1, Mar. 1964, pp. 5973.
KLIMKO, E. and M. F. Neuts, "The Single Server Queue in Discrete Time Numerical Analysis II," Vol. 20, No. 2, June 1973,
pp. 305319.
KLIMKO, E. and M. F. Neuts, "The Single Server Queue in Discrete Time Numerical Analysis III," Vol. 20, No. 3, Sept. 1973,
pp. 557567.
CUMULATIVE INDEX FOR VOLUMES 120
827
KLINGMAN, D., A. Charnes and F. Glover, "The Lower Bounded and Partial Upper Bounded Distribution Model," Vol. 18,
No. 2, June 1971, pp. 277281.
KOLESAR, P. J., "Linear Programming and the Reliability of MultiComponent Systems," Vol. 14, No. 3, Sept. 1967, pp. 317
327.
KOOPMANS, L. H., "A Statistical Study of the Derailment Hazard for U.S. Class I Railways," Vol. 12, No. 1, Mar. 1965, pp.
95118.
KORTANEK, K. O., "MultiProduct Production Scheduling Via Extreme Point Properties of Linear Programming," Vol. 15, No.
2, June 1968, pp. 287300.
KORTANEK, K. O. and S. A. Gustafson, "Numerical Treatment of a Class of SemiInfinite Programming," Vol 20, No. 3,
Sept. 1973, pp. 477504.
KORTANEK, K. O., A Charnes and W. W. Cooper, "On the Theory of SemiInfinite Programming and a Generalization of the
KuhnTucker Saddle Point Theorem for Arbitrary Convex Functions," VoL 16, No. 1 , Mar. 1969, pp. 4151.
KRIEBEL, C. H., "Team Decision Models of an Inventory Supply Organization," Vol. 12, No. 2, June 1965, pp. 139154.
KRUSKAL, J. B. and R. J. Aumann, "Assigning Quantitative Values to Qualitative Factors in the Naval Electronics Problem,"
Vol. 6, No. 1, Mar. 1959, pp. 116.
KRUSKAL, J. B. and R. J. Aumann, "The Coefficients in an Allocation Problem," VoL 5, No. 2, June 1958, pp. 111123.
KUHN, H. W., "The Hungarian Method for the Assignment Problem," Vol. 2, Nos. 1 & 2, Mar.June 1955, pp. 8397.
KUHN, H. W., "Variants of the Hungarian Method for Assignment Problems," Vol. 3, No. 4, Dec. 1956, pp. 253258.
KUHN, H. W. and W. J. Baumol, "An Approximative Algorithm for the FixedCharges Transportation Problem," Vol. 9, No. 1,
Mar. 1962, pp. 115.
KYDLAND, F., "Duality in Fractional Programming," VoL 19, No. 4, Dec. 1972, pp. 691697.
LADERMAN, J. and M. Beckmann, "A Bound on the Use of Inefficient Indivisible Units," Vol. 3, No. 4, Dec. 1956, pp. 245252.
LADERMAN, J., L. Gleiberman and J. F. Egan, "Vessel Allocation by Linear Programming," Vol. 13, No. 3, September 1966,
pp. 315320.
LAGEMANN, J. J., "A Method for Solving the Transportation Problem," Vol. 14, No. l,Mar. 1967, pp. 8999.
LANGFORD, E., "A Continuous Submarine Versus Submarine Game," Vol. 20, No. 3, Sep. 1973, pp. 405417.
LASDON, L. S. and J. K. Hartman, "A Generalized Upper Bounding Method for Doubly Coupled Linear Programs," Vol. 17,
No. 4, Dec. 1970, pp. 411^129.
LEDLEY, R. S. and L. S. Rotolo, "A Heuristic Concept and an Automatic Computer Program Aid for Operational Simulation,"
VoL 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 231244.
LEFKOWITZ, B. and D. A. D'Esopo, "Note on a Linear Programming Model for Determining a Minimum Embarkation Fleet,"
Vol. 11, No. 1, Mar. 1964, pp. 7982.
LEFKOWITZ, B., D. A. D'Esopo and H. L. Dixion, "A Model for Simulating an AirTransportation System," Vol. 7, No. 3, Sept.
1960, pp. 213220.
LEMKE, C. E., "The Dual Method of Solving the Linear Programming Problem," Vol. 1, No. 1, Mar. 1954, pp. 36^17.
LEMKE, C. E. and A. Charnes, "Minimization of NonLinear Separable Convex Functionals," Vol. 1, No. 4, Dec. 1954, pp.
301312.
LENEMAN, O. A. Z. and F. J. Beutler, "On a New Approach to the Analysis of Stationary Inventory Problems," Vol. 16, No. 1,
Mar. 1969, pp. 115.
LEVIN, O. and Y. Almogy, "The Fractional FixedCharge Problem," Vol. 18, No. 3, Sept. 1971, pp. 307315.
LEVITAN, R. E., "Toward a Study of Bidding Processes: Part IV Games with Unknown Costs," Vol. 14, No. 4, Dec. 1967,
pp. 415433.
LEVY, E. and D. R. Limaye, "A Probabilistic Evaluation of Helicopter Lift Capability," Vol. 19, No. 4, Dec. 1972, pp. 761775.
LEVY, F. K., G. L. Thompson and J. D. Wiest, "Multiship, Multishop, WorkloadSmoothing Program," Vol. 9, No. 1, Mar. 1962,
pp. 37^4.
LEVY, J., "Further Notes on the Loss Resulting from the Use of Incorrect Data in Computing an Optimal Inventory Policy,"
VoL 6, No. 1, Mar. 1959, pp. 2531.
LEVY, J., "Loss Resulting from the Use of Incorrect Data in Computing an Optimal Inventory Policy," Vol. 5, No. 1, Mar. 1958,
pp. 7581.
LEWIS, R., M. Gourary and F. Neeland, "An Inventory Control Bibliography," Vol. 3, No. 4, Dec. 1956, pp. 295305.
LEWIS, R. W., E. R. Rosholdt and W. Wilkinson, "A MultiMode Transportation Network Model," Vol. 12, Nos. 3 & 4, Sept.Dec.
1965, pp. 261274.
LIEBERMAN, G. J., "The Status and Impact of Reliability Methodology," Vol. 16, No. 1, Mar. 1969, pp. 1735.
LIEBERMAN, G. J., C. Derman and S. M. Ross, "On Optimal Assembly of Systems," Vol. 19, No. 4, Dec. 1972, pp. 569574.
LIEBMAN, J. C. and N. Dee, "Optimal Location of Public Facilities," Vol. 19, No. 4, Dec. 1972, pp. 753759.
LIEBMAN, J. C, M. Bellmore and D. H. Marks, "An Extension of the (Szwarc) Truck Assignment Problem," Vol. 19, No. 1,
Mar. 1972, pp. 9199.
LIENTZ, B. and R. Alter, "A Note on a Problem of Smirnov: A Graph Theoretic Interpretation," Vol. 17, No. 3, Sept. 1970,
pp. 407^08.
LIENTZ, B. P. and R. Alter, "Applications of a Generalized Combinatorial Problem of Smirnov," Vol. 16, No. 4, Dec. 1969,
pp. 543547.
828 CUMULATIVE INDEX FOR VOLUMES 120
LIMAYE. D. R. and E. B. Brandy, "MAD: Mathematical Analysis of Downtime," Vol. 17, No. 4, Dec. 1970, pp. 525534.
LIMAYE, D. R. and E. Levy, "A Probabilistic Evaluation of Helicopter Lift Capability," Vol. 19, No. 4, Dec. 1972, 761775.
LINDEMAN, A. R., V. L. Smith and R. Saposnik, "Allocation of a Resource to Alternative Probabilistic Demands: Transport
Equipment Pool Assignments," Vol. 6, No. 3, Sept. 1959, pp. 193207.
LIPPMAN, S. A. and J. S. C. Yuan, "Discounted Production Scheduling and Employment Smoothing," Vol. 16, No. 1, Mar.
1969, pp. 93110.
LOCKS, M. O., "A Bivariate Normal Theory MaximumLikelihood Technique when Certain Variances Are Known," Vol. 18,
No. 4, Dec. 1971, pp. 525530.
LOMB ARDI, L.. " A Direct Method for the Computation of Waiting Times," Vol. 8, No. 2, June 1961 , pp. 193197.
LONGHILL, J. D. and J. Bracken, "Note on a Model for Minimizing Cost of Aerial Tanker Support of a Practice Bomber Mis
sion," Vol. 11, No. 4, Dec. 1964, pp. 359364.
LOVE, R. F., "Locating Facilities in ThreeDimensional Space by Convex Programming," Vol. 16, No. 4, Dec. 1969, pp. 503
516.
LOVE, R. F. and G. O. Wesolowsky, "Location of Facilities with Rectangular Distances Among Point and Area Destinations,"
Vol. 18, No. 1, Mar. 1971, pp. 8390.
LU. J. Y. and R. B. S. Brooks, "War Reserve Spares Kits Supplemented by Normal Operating Assets," Vol. 16, No. 2, June
1969, pp. 229236.
LUBORE, S., "A Maximum Utility Solution to a Vehicle Constrained Tanker Scheduling Problem," Vol. 15, No. 3, Sept. 1968,
pp. 403412.
LUBORE, S., G. Bennington and L. Gsellman, "An Economic Model for Planning Strategic Mobility Posture," Vol. 19, No. 3,
Sept. 1972, pp. 461470.
LUBORE, S. H., H. D. RatlifF and G. T. Sicilia, "Determining the Most Vital Link in a Flow Network," Vol. 18, No. 4, Dec. 1971,
pp. 497502.
LUBORE. S.. M. Bellmore and G. Bennington, "A Network Isolation Algorithm," Vol. 17, No. 4, Dec. 1970, pp. 461469.
LUBORE, S. and G. Bennington, "Resource Allocation for Transportation," Vol. 17, No. 4, Dec. 1970, pp. 471484.
LUCAS. W. F., "Solutions for a Class of nPerson Games in Partition Function Form," Vol. 14, No. 1, Mar. 1967, pp. 1521.
LUCAS, W. F. and R. M. Thrall, "nPerson Games in Partition Function Form," Vol. 10, No. 4, Dec. 1963, pp. 281298.
LUTFUL KABIR, A. B. M. and L. K. Chan, "Optimum Quantiles for the Linear Estimation of the Parameters of the Extreme
Value Distribution in Complete and Censored Samples," Vol. 16, No. 3, Sept. 1969, pp. 381404.
LUTHER. M. and E. Walsh, "A Difficulty in Linear Programming for Transportation Problems," Vol. 5, No. 4, Dec. 1958, pp.
363366.
LYNCH, C. F.. "Notes on Applied Analytical Logistics in the Navy," Vol. 1, No. 2, June 1954, pp. 90102.
MACKENZIE, D. C, "Contractor Performance," Vol. 2, Nos. 1 & 2, Mar. June 1955, pp. 1723.
MACQUEEN, J. B. and D. F. Hitchcock, "On Computing the Expected Discounted Return in a Markov Chain," Vol. 17, No. 2,
June 1970, pp. 237241.
MADOW, W. G., S. Karlin and W. E. Pruitt, "On Choosing Combinations of Weapons," Vol. 10, No. 2, June 1963, pp. 95119.
MAGAZINE, M. J.. "Optimal Control of MultiChannel Service Systems," Vol. 18, No. 2, June 1971, pp. 177183.
MAGRUDER, C. B., J. Bracken, C. B. Brossman and A. D. Tholen, "A Theater Materiel Model," Vol. 12, Nos. 3 & 4, Sept.Dec.
1965, pp. 295313.
MALIK, H. J., "A Note on Generalized Inverses," Vol. 15, No. 3, Sept. 1968, pp. 605612.
MALIK, H. J.. "The Distribution of the Product of Two NonCentral Beta Variates," Vol. 17, No. 3, Sept. 1970, pp. 327330.
MANGASARIAN,0. L, "Equivalence in Nonlinear Programming," Vol. 10, No. 4, Dec. 1963, pp. 299306.
MANN, N. R., "ComputerAided Selection of Prior Distributions for Generating Monte Carlo Confidence Bounds on System
Reliability," Vol. 17, No. 1, Mar. 1970, pp. 4154.
MANN, N. R., "A Test for the Hypothesis That Two ExtremeValue Scale Parameters Are Equal," Vol. 16, No. 2, June 1969,
pp. 207216.
MANNE, A. S., "Comments on interindustry Economics' by Chenery and Clark," Vol 7, No. 4, Dec. 1960, pp. 385389.
MANNE, A. S., "On the Timing of Development Expenditures and the Retirement of Military Equipment," Vol. 8, No. 3, Sept.
1961, pp. 235243.
MARCHI, E., "Simple Stability of General nPerson Games," Vol. 14, No. 2, June 1967, pp. 163171.
MARKLAND, R. E., "A Comparative Study of Demand Forecasting Techniques for Military Helicopter Spare Parts," Vol. 17,
No. 1, Mar. 1970, pp. 103119.
MARKOWITZ, H. M., "Computing Procedures for Portfolio Selection (Abstract), "Vol. 4, No. 1, Mar. 1957, pp. 8788.
MARKOWITZ, H. M., "The Optimization of a Quadratic Function Subject to Linear Constraints," Vol. 3, Nos. 1 & 2, Mar. June
1956, pp. 111133.
MARKOWITZ. H. M. and A. J. Hoffman, "Shortest Path Assignment and Transportation Problems," Vol. 10, No. 4, Dec. 1963,
pp. 375379.
CUMULATIVE INDEX FOR VOLUMES 120 829
MARKS, D. H., M. Bellmore and J. C. Liebman, "An Extension of the (Szwarc) Truck Assignment Problem," Vol. 19, No. 1,
Mar. 1972, pp. 9199.
MARLOW, W. H., "Some Accomplishments of Logistics Research." Vol. 7, No. 4, Dec. 1960, pp. 299314.
MARLOW, W. H., M. Denicoff, J. Fennell, S. E. Haber, F. W. Segel and H. Solomon, "The Polaris Military Essentiality System,"
Vol. 11, No. 4, Dec. 1964, pp. 235257.
MARLOW, W. H., M. Denicoff, J. Fennell, S. E. Haber and H. Solomon, "A Polaris Logistics Model," Vol. 11, No. 4, Dec. 1964,
pp. 259272.
MARLOW, W. H. and J. R. Isbell, "Attrition Games," Vol. 3, Nos. 1 & 2, Mar. June 1956, pp. 7194.
MARSCHAK, J. and M. R. Mickey, "Optimal Weapon Systems," Vol. 1. No. 2, June 1954, pp. 116140.
MARSHALL, C. W., "Quantification of Contractor Risk," Vol. 16, No. 4, Dec. 1969, pp. 531541.
MARSHALL, K. T. and J. W. Suurballe, "A Note on Cycling in the Simplex Method," Vol. 16, No. 1, Mar. 1969, pp. 121137.
MARTIN, J. L, JR., "Economic Impact and the Notion of Compensated Procurement," Vol. 15, No. 1, Mar. 1968, pp. 6379.
MARTOS, B., "Hyperbolic Programming," Vol. 11, Nos. 2 & 3, JuneSept. 1964. pp. 135155.
MARTZ, H. F. and G. K. Bennett, "An Empirical Bayes Estimator for the Scale Parameter of the TwoParameter Weibull Dis
tribution," Vol. 20, No. 3, Sept. 1973, pp. 387393.
MASCHLER, M., "A Price Leadership Method for Solving the Inspector's NonConstantSum Game," Vol. 13, No. 1, Mar.
1966, pp. 1133.
MASCHLER, M., "The Inspector's NonConstantSum Game: Its Dependence on a System of Detectors," Vol. 14, No. 3, Sept.
1967, pp. 275290.
MASCHLER, M. and M. Davis, "The Kernel of a Cooperative Game," Vol. 12. Nos. 3 & 4, Sept.Dec. 1965, pp. 223259.
MASTRAN, D. V. and C. J. Thomas, "Decision Rules for Attacking Targets of Opportunity," Vol. 20, No. 4, Dec. 1973, pp.
661672.
MASTROBERTI, R. and R. L. Nolan, "Productivity Estimates of the Strategic Airlift System by the Use of Simulation," Vol.
19, No. 4, Dec. 1972, pp. 737752.
MAXWELL, W. L. and T. B. Crabill, "Single Machine Sequencing with Random Processing Times and Random DueDates,"
Vol. 16, No. 4, Dec. 1969, pp. 549554.
MAXWELL, W. L., "MultipleFactor Rules for Sequencing with Assembly Constraints," Vol. 15, No. 2, June 1968, pp. 241254.
MAXWELL, W. L., "The Scheduling of Economic Lot Sizes," Vol. 11, Nos. 2 & 3, JuneSept. 1964, pp. 89124.
MAZUMDAR, M., "Statistical Estimation in a Problem of System Reliability," Vol. 14, No. 4, Dec. 1967. pp. 473488.
MAZUMDAR, M., "Uniformly Minimum Variance Unbiased Estimates of Operational Readiness and Reliability in a TwoState
System," Vol. 16, No. 2, June 1969, pp. 199206.
MAZUMDAR, M., "Some Estimates of Reliability Using Interference Theory." Vol. 17, No. 2, June 1970, pp. 159165.
MCCLOSKEY, J. F. and F. Hanssman, "An Analysis of Stewardess Requirements and Scheduling for a Major Domestic Air
line," Vol. 4, No. 3, Sept. 1957, pp. 183192.
MCMASTERS, A. W. and T. M. Mustin, "Optimal Interdiction of a Supply Network," Vol. 17, No. 3, Sept. 1970, pp. 261268
MCSHANE, R. E., "Science and Logistics," Vol. 2, Nos. 1 & 2, Mar. June, 1955, pp. 17.
MEAD, E. R., L. K. Chan and S. W. H. Cheng, "An Optimum tTest for the Scale Parameter of an ExtremeValue Distribution,'
Vol. 19. No. 4, Dec. 1972, pp. 715723.
MEADE, R., JR. and C. A. Fischer, "Mobile Logistics Support in the 'Passage to Freedom' Operation," Vol. 1, No. 4, Dec. 1954
pp. 258264.
MEANS, E. H., "Relationships Among Potential Sorties, Ground Support, and Aircraft Reliability," Vol. 15, No. 4, Dec. 1968
pp. 491506.
MEHRA, M., "MultipleFactor Rules for Sequencing with Assembly Constraints," Vol. 15, No. 2, June 1968, pp. 241254.
MEISNER, M., "Cannibalization in Multicomponent Systems and the Theory of Reliability," Vol. 15, No. 3, Sept. 1968, pp
331360.
MELLON, W., "A Selected Descriptive Bibliography of References on Priority Systems and Related, NonPrice Allocators,'
Vol. 5, No. 1, Mar. 1958, pp. 1727.
MELLON, W. G., "Priority Ratings in More Than One Dimension," Vol. 7, No. 4, Dec. 1960, pp. 513527.
MENON, V. V., "The Minimal Cost Flow Problem with Convex Costs," Vol. 12, No. 2, June 1965, pp. 163172.
MERTEN, A. G. and K. R. Baker, "Scheduling with Parallel Precessors and Linear Delay Costs," Vol. 20, No. 4, Dec. 1973, pp.
793804.
MEYER, R. F. and H. B. Wolfe, "The Organization and Operation of a Taxi Fleet," Vol. 8, No. 2, June 1961, pp. 137150.
MIKHEY, M. R., "A Method for Determining Supply Quantity for the Case of Poisson Distribution of Demand," Vol. 6, No. 4,
Dec. 1959, pp. 265272.
MICKEY, M. R. and J. Marschak, "Optimal Weapon Systems," Vol. 1, No. 2, June 1954, pp. 116140.
MILLER, M. H. and A. Charnes, "Mathematical Programming and Evaluation of Freight Shipment Systems, Application and
Analysis," Part II, "Analysis," Vol. 4, No. 3, Sept. 1957, pp. 243252.
MILLER, M., A. Charnes, and W. W. Cooper, "Dyadic Programs and Subdual Methods," Vol. 8, No. 1, Mar. 1961, pp. 123.
MILLHAM, C. B., "Constructing Bimatrix Games with Special Properties," Vol. 19, No. 4, Dec. 1972. pp. 709714.
MILLS, H. D., "Organized Decision Making," Vol. 2, No. 3, Sept. 1955, pp. 137143.
830 CUMULATIVE INDEX FOR VOLUMES 120
MITCHELL, W. E. and R. L. Disney, "A Solution for Queues with Instantaneous Jockeying and Other Customer Selection
Rules," Vol. 17, No. 3, Sept. 1970, pp. 315325.
MOESCHLIN,0. and G. Bol, "Applications of Mills' Differential," Vol. 20, No. l,Mar. 1973, pp. 101108.
MOGLEWER, S. and C. Payne, "A Game Theory Approach to Logistics Allocation," Vol. 17, No. 1, Mar. 1970, pp. 8797.
MOHAN, C. and K. L. Arora, "Analytical Study of a Problem in Air Defense," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 275
279.
MOHAN, C. and R. C. Garg, "Decision on Retention of Excess Stock Following a Normal Probability Law of Obsolescence and
Deterioration," Vol. 8, No. 3, Sept. 1961, pp. 229234.
MOND, B., "On the Direct Sum and Tensor Product of Matrix Games," Vol. 11, Nos. 2 and 3, JuneSept. 1964, pp. 205215.
MOND, B. and B. D. Craven, "A Note on Mathematical Programming with Fractional Objective Functions," Vol. 20, No. 3,
Sept. 1973, pp. 577581.
MONTGOMERY, D. C, M. S. Bazaraa and A. K. Keswani, "Inventory Models with a Mixture of Backorders and Lost Sales,"
Vol. 20, No. 2, June 1973, pp. 255263.
MONTGOMERY, D. C, P. M. Ghare and W. C. Turner, "Optimal Interdiction Policy for a Flow Network," Vol. 18, No. 1,
Mar. 1971, pp. 3745.
MOREY, R. C. and D. L. Iglehart, "Optimal Policies for a MultiEchelon Inventory System with Demand Forecasts," Vol. 18,
No. l,Mar. 1971, pp. 115118.
MOREY, R. C, "Inventory Systems with Imperfect Demand Information," Vol. 17, No. 3, Sept. 1970, pp. 287295.
MORGENSTERN, O., "Consistency Problems in the Military Supply System," Vol. 1, No. 4, Dec. 1954, pp. 265281.
MORGENSTERN, O., "Note on the Formulation of the Theory of Logistics," Vol. 2, No. 3, Sept. 1955, pp. 129136.
MORGENSTERN, O. and G. L. Thompson, "A Note on an Open Expanding Economy Model," Vol. 19, No. 3, Sept. 1972, pp.
557559.
MORRILL, J. E., "OnePerson Games of Economic Survival," Vol. 13, No. 1, Mar. 1966, pp. 4969.
MORTON, T. E. and S. P. Sethi, "A Mixed Optimization Technique for the Generalized Machine Replacement Problem,"
Vol. 19, No. 3, Sept. 1972, pp. 471481.
MUKHOPADHYAY, A. C. and T. S. Arthanari, "A Note on a Paper by W. Szwarc," Vol. 18, No. 1, Mar. 1971, pp. 135138.
MUNKRES, J., "On the Assignment and Transportation Problems (Abstract)," Vol. 4, No. 1, Mar. 1957, pp. 7778.
MURPHY, R. E., F. D. Dorey and R. D. Campbell, "Concept of a Logistics System," Vol. 4, No. 2, June 1957, pp. 101116.
MUSTIN, T. M. and A. W. McMasters, "Optimal Interdiction of A Supply Network," Vol. 17, No. 3, Sept. 1970, pp. 261268.
NAHMIAS, S. and W. P. Pierskalla, "Optimal Ordering Policies for a Product that Perishes in Two Periods Subject to Sto
chastic Demand," Vol. 20, No. 2, Jun. 1973, pp. 207229.
NAIR, K. P. K. and R. Chandrasekaran, "Optimal Location of a Single Service Center of Certain Types," Vol. 18, No. 4, Dec.
1971, pp. 503510.
NAOR, P., "On Queueing Systems with Variable Service Capacities," Vol. 14, No. 1, Mar. 1967, pp. 4353.
NEELAND, F., R. Lewis and M. Gourary, "An Inventory Control Bibliograph, " Vol. 3, No. 4, Dec. 1956, pp. 295303.
NELSON, R. T., "Queueing Network Experiments with Varying Arrival ana 1 Service Processes," Vol. 13, No. 3, Sept. 1966,
pp. 321347.
NEMHAUSER, G. L., "Computational Results for a Stopping Rule Problem on Averages," Vol. 15, No. 4, Dec. 1968, pp. 567578.
NEMHAUSER, G. L., "Decomposition of Linear Programs by Dynamic Programming," Vol. 11, Nos. 2 and 3, JuneSept. 1964,
pp. 191195.
NEMHAUSER, G. L., "Optimal Capacity Expansion," Vol. 15, No. 4, Dec. 1968, pp. 531550.
NEMHAUSER, G. L., M. Bellmore and W. D. Eklof, "A Decomposable Transshipment Algorithm for a Multiperiod Trans
portation Problem," Vol. 16, No. 4, Dec. 1969, pp. 517524.
NEMHAUSER, G. L. and V. J. Bowman, Jr., "A Finiteness Proof for Modified Dantzig Cuts in Integer Programming," Vol. 17,
No. 3, Sept. 1970, pp. 309313.
NEUTS, M. F.. "The Single Server Queue in Discrete TimeNumerical Analysis I," Vol. 20, No. 2, Jun. 1973, pp. 297304.
NEUTS, M. F. and D. Heimann, "The Single Server Queue in Discrete TimeNumerical Analysis IV," Vol. 20, No. 4, Dec. 1973,
pp. 753766.
NEUTS, M. F. and E. Klimko, "The Single Server Queue in Discrete TimeNumerical Analysis II," Vol. 20, No. 2, June 1973.
pp. 305319.
NEUTS, M. F. and E. Klimko, "The Single Server Queue in Discrete TimeNumerical Analysis III," Vol. 20, No. 3, Sept. 1973,
pp. 557567.
NIGHTENGALE, M. E., "The Value Statement," Vol. 17, No. 4, Dec. 1970, pp. 507514.
NIKOLAISEN, T. and R. W. Butterworth, "Bounds on the Availability Function," Vol. 20, No. 2, Jun, 1973, pp. 289296.
NOBLE, S. B., "Some Flow Models of Production Constraints," Vol. 7, No. 4, Dec. 1960, pp. 401419.
NOBLE, S. B. and M. Hershkowitz, "Finding the Inverse and Connections of a Type of Large Sparse Matrix," Vol. 12, No. 1,
Mar. 1965, pp. 119132.
NOLAN, R. L., "Systems Analysis and PlanningProgrammingBudgeting Systems (PPBS) for Defense Decision Making,"
Vol. 17, No. 3, Sept. 1970, pp. 359372.
NOLAN, R. L. and R. Mastroberti, "Productivity Estimates of the Strategic Airlift System by the Use of Simulation,"
Vol. 19, No. 4, Dec. 1972, pp. 737752.
O'BRIEN, M. J., "Programming the Procurement of Airlift and Sealift Forces: A Linear Programming Model for Analysis of
the LeastCost Mix of Strategic Deployment Systems," Vol. 14, No. 2, June 1967, pp. 241255.
CUMULATIVE INDEX FOR VOLUMES 120 831
OKUN, B. , "Design, Test, and Evaluation of an Experimental Flyaway Kit," Vol. 7, No. 2, June 1960, pp. 109136.
O'NEILL, R. R., "Analysis and Monte Carlo Simulation of Cargo Handling," Vol. 4, No. 3, Sept. 1957, pp. 223236.
O'NEILL, R. R. , "Scheduling of Cargo Containers," Vol. 7, No. 4, Dec. 1960, pp. 577584.
O'NEILL, R. R. and J. K. Weinstock, "The CargoHandling System" Vol. 1 , No. 4, Dec. 1954, pp. 282288.
ORCHARDHAYS, W., "Control of and Communication with DataHandling Machines," Vol. 7, No. 4, Dec. 1960, pp. 357363.
OSHIRO, S. and J. Fennel, "The Dynamics of Overhaul and Replenishment Systems for Large Equipments," Vol. 3, Nos.
1 & 2, Mar. June 1956, pp. 1943.
OWEN, D. B., "An Application of Statistical Techniques to Estimate Engineering ManHours on Major Aircraft Programs,"
Vol. 15, No. 4, Dec. 1968, pp. 579589.
OWEN, G., "Political Games," Vol. 18, No. 3, Sept. 1971, pp. 345355.
PADBERG, M. W., "Equivalent KnapsackType Formulations of Bounded Integer Linear Programs: An Alternative Approach,"
Vol. 19, No. 4, Dec. 1972, pp. 699708.
PANWALKAR, S. S., "Parametric Analysis of Linear Programs with Upper Bounded Variables," Vol. 20, No. 1, Mar. 1973,
pp. 8393.
PARKER, J. B., "Bayesian Prior Distributions for MultiComponent Systems," Vol. 19, No. 3, Sept. 1972, pp. 509515.
PARKER, L. L., "Economical ReOrder Quantities and ReOrder Points with Uncertain Demand," Vol. 11, No. 4, Dec. 1964,
pp. 351358.
PATTERSON, J. H., "Alternate Methods of Project Scheduling with Limited Resources," Vol. 20, No. 4, Dec. 1973, pp. 767784.
PAULSEN, M., W. Hall and H. Bremer, "Experiences with the Bid Evaluation Problem (Abstract)," Vol. 4, No. 1, Mar. 1957,
p. 27.
PAYNE, C. and S. Moglewer, "A Game Theory Approach to Logistics Allocation," Vol. 17, No. 1, Mar. 1970, pp. 8797.
PELEG, B., "An Inductive Method for Constructing Minimal Balanced Collections of Finite Sets," Vol. 12, No. 2, June 1965,
pp. 155162.
PELEG, B., "Utility Functions of Money for Clear Games," Vol. 12, No. 1 , Mar. 1965, pp. 5764.
PENNINGTON, A. W., E. W. Rice and J. Bracken, "Allocation of CarrierBased Attack Aircraft Using NonLinear Program
ming," Vol. 18, No. 3, Sept. 1971, pp. 379393.
PERLAS, M., J. M. Burt, Jr., and D. P. Gaver, "Simple Stochastic Networks: Some Problems and Procedures," Vol. 17, No. 4,
Dec. 1970, pp. 439459.
PERSINGER, C. A., "Optimal Search Using Two Nonconcurrent Sensors," Vol. 20, No. 2, Jun. 1973, pp. 277288.
PETERSEN, J. W. and M. A. Geisler, "The Costs of Alternative Air Base Stocking and Requisitioning Policies," Vol. 2, Nos.
1 & 2, Mar. June 1955, pp. 6981.
PETERSEN, J. W. and W. A. Steger, "Design Change Impacts on Airframe Parts Inventories," Vol. 5, No. 3, Sept. 1958, pp.
241255.
PFANZAGL, J., "A General Theory of Measurement Applications to Utility," Vol. 6, No. 4, Dec. 1959, pp. 283294.
PFOUTS, R. W., "A Note on Systems of Simultaneous Linear Difference Equations with Constant Coefficients," Vol. 12, Nos.
3 & 4, Sept.Dec. 1965, pp. 335340.
PIERCE, D. A., "Computational Results for a Stopping Rule Problem on Averages," Vol. 15, No. 4, Dec. 1968, pp. 567578.
PIERCE, J. F. and W. B. Crowston, "TreeSearch Algorithms for Quadratic Assignment Problems," Vol. 18, No. 1, Mar. 1971,
pp. 136.
PIERSKALLA, W. P. and S. Nahmias, "Optimal Ordering Policies for a Product that Perishes in Two Periods Subject to Sto
chastic Demand," Vol. 20, No. 2, Jun. 1973, pp. 207229.
POLLACK, M., "Some Studies on Shuttle and AssemblyLine Processes," Vol. 5, No. 2, June 1958, pp. 125136.
POLLACK, S. M., "Allocation of Resources to Randomly Occurring Opportunities," Vol. 14, No. 4, Dec. 1967, pp. 513527.
PORTER, D. B., "The Gantt Chart as Applied to Production Scheduling and Control," Vol. 15, No. 2, June 1968, pp. 311318.
POSNER, M. J. M. and B. Yansouni, "A Class of Inventory Models with Customer Impatience," Vol. 19, No. 3, Sept. 1972,
pp. 483492.
PRAGER, W., "Numerical Solution of the Generalized Transportation Problem," Vol. 4, No. 3, Sept. 1957, pp. 253261.
PRAGER,W. andT. C. Hu, "Network Analysis of Production Smoothing," Vol. 6, No. l,Mar. 1959, pp. 1723.
PRAWDA, J., "ProductionAllocation Scheduling and Capacity Expansion Using Network Flows under Uncertainty," Vol. 20,
No. 3, Sept. 1973, pp. 517531.
PRESUTTI, V. J., Jr., and R. C. Trepp, "More Ado About Economic Order Quantities (EOQ)," Vol. 17, No. 2, June 1970, pp.
243251.
PROSCHAN, F., "Optimal System Supply," Vol. 7, No. 2, Dec. 1960, pp. 609646.
PROSCHAN, F. and L. Hunter, "Replacement When Constant Failure Rate Precedes Wearout," Vol. 8, No. 2, June 1961,
pp. 127136.
PRUITT, W. E., "A Class of Dynamic Games," Vol. 8, No. 1, Mar. 1961, pp. 5578.
PRUITT, W. E., S. Karlin and W. G Madow, "On Choosing Combinations of Weapons," Vol. 10, No. 2, June 1963, pp. 95
119.
PRUZAN, P. M., "The ManyProduct Cargo Loading Problem," Vol. 14, No. 3, Sept. 1967, pp. 381390.
QUANDT, R. E., "On the Solution of Probabilistic Leontief Systems," Vol. 6, No. 4, Dec. 1959, pp. 295305.
QUANDT, R. E., "Probabilistic Errors in the Leontief System," Vol. 5, No. 2, June 1958, pp. 155170.
832 CUMULATIVE INDEX FOR VOLUMES 120
RAGO, L. J., "Sequencing, Modeling, and Gantt Charting Repetitive Manufacturing," Vol. 15, No. 2, June 1968, pp. 301310.
RAHIM, M. A. and M. Ahsanullah, "Simplified Estimates of the Parameters of the Double Exponential Distribution Based on
Optimum Order Statistics from a MiddleCensored Sample," Vol. 20, No. 4, Dec. 1973, pp. 745751.
RAMSEY, F. A., Jr., "Damage Assessment Systems and Their Relationship to PostNuclearAttack Damage and Recoverv "
Vol. 5, No. 3, Sept. 1958, pp. 199219.
RANDOLPH, P. H. and G. E. Swinson, "The Discrete MaxMin Problem," Vol. 16, No. 3, Sept. 1969, pp. 309314.
RAO, M. R. and R. S. Garfinkel, "The Bottleneck Transportation Problem," Vol. 18, No. 4. Dec. 1971, pp. 465472.
RAO, M. R., R. Jagannathan and W. Dent. "Parametric Linear Programming: Some Special Cases," Vol. 20, No. 4, Dec. 1973,
pp. 725728.
RASOF, B. and L. S. Abrams, "A 'Static' Solution to a 'Dynamic' Problem in Acquisition Probability," Vol. 12, No. 1, Mar. 1965,
pp. 6594.
RATLIFF, H. D., S. H. Lubore and G. T. Sicilia, "Determining the Most Vital Link in a Flow Network," Vol. 18, No. 4, Dec. 1971,
pp. 497502.
RAU, J. G., "A Model for Manpower Productivity during Organization Growth," Vol. 18, No. 4, Dec. 1971, pp. 543559.
RAVINDRAN, A., "Optimal Inventory Policies in Contagious Demand Models," Vol. 19, No. 1, Mar. 1972, pp. 191203.
RAVINDRAN, A., "A Comparison of the PrimalSimplex and Complementary Pivot Methods for Linear Programming," Vol. 20,
No. 1, Mar. 1973. pp. 95100.
RAY, T. L. and P. L. Davis. "A BranchBound Algorithm for the Capacitated Facilities Location Problem," Vol. 16, No. 3, Sept.
1969, pp. 331344.
READ, R. R., "On the Output of Parallel Exponential Service Channels." Vol. 16, No. 4, Dec. 1969, pp. 555572.
REED, R. and J. Hale, "A Formulation of the Decision Problem for a Class of Systems." Vol. 3. No. 4. Dec. 1956, pp. 259277.
REYNOLDS, D. F., "An Application of Statistical Techniques to Estimate Engineering ManHours on Major Aircraft Programs,"
Vol. 15. No. 4, Dec. 1968, pp. 579589.
RHODE, A. S., J. J. Gelke and F. X. Cook, "Impact of an All Volunteer Force upon the Navy in the 197273 Timeframe," Vol. 19,
No. 1, Mar. 1972, pp. 4375.
RICE, E. W., J. Bracken and A. W. Pennington, "Allocation of CarrierBased Attack Aircraft Using NonLinear Programming,"
Vol. 18, No. 3, Sept. 1971, pp. 379393.
RICHARDSON, H. R. and L. D. Stone, "Operations Analysis During the Underwater Search for Scorpion" Vol. 18, No. 2,
Jun. 1971, pp. 141157.
RICHARDSON, M. and F. Harary, "A Matrix Algorithm for Solutions and rBases of a Finite Irreflexive Relation," Vol. 6, No. 4,
Dec. 1959, pp. 307314.
RIGBY, F. D., "An Analog and Derived Algorithm for the Dual Transportation Problem," Vol. 9, No. 2, June 1962, pp. 8196.
RIGBY, F. F., Introductory Note to the Paper "Equilibrium Points in Games with Vector Payoffs," by Shapley, L. S., Vol. 6,
No. LMar. 1959, pp. 5761.
RITTER, K., "A Method for Solving Nonlinear MaximumProblems Depending on Parameters," Vol. 14, No. 2, June 1967, pp.
147162.
ROBBINS, J. J., Translation of the paper "Tank Duel with Game Theory Implications," by Zachrisson, L. E., Vol. 4, No. 2, June
1957, pp. 131138.
ROBERTS, S. and H. Heck, "A Note on the Extension of a Result on Scheduling with Secondary Criteria," Vol. 19, No. 2, Jun.
1972, pp. 403405.
ROBILLARD, P., "(0,1) Hyperbolic Programming Problems," Vol. 18, No. l,Mar. 1971, pp. 4757.
ROBILLARD, P., P. Bratley and M. Florian, "Scheduling with Earliest Start and Due Date Constraints," Vol. 18, No. 4, Dec.
1971, pp. 511519.
ROBILLARD, P., M. Florian and P. Bratley, "On Sequencing with Earliest Starts and Due Dates with Application to Computing
Bounds for the (nlmlGIF max ) Problem," Vol. 20, No. 1, Mar. 1973, pp. 5767.
ROELOFFS, R., "Minimax Surveillance Schedules for Replaceable Units," Vol. 14, No. 4, Dec. 1967, pp. 461471.
ROELOFFS. R., "Minimax Surveillance Schedules with Partial Information," Vol. 10, No. 4, Dec. 1963, pp. 307322.
ROGERS, W. F. and G. F. Brown, "A Bayesian Approach to Demand Estimation and Inventory Provisioning," Vol. 20, No. 4,
Dec. 1973, pp. 607624.
ROLFE, A. J., "Markov Chain Analysis of a Situation Where Cannibalization is the Only Repair Activity," Vol. 17, No. 2, June
1970, pp. 151158.
ROLLOF,Y., "System Analysis and/or Common Sense," Vol. 3, Nos. 1 & 2, Mar. June 1956, pp. 1118.
ROODMAN, G. M., "Postoptimality Analysis in ZeroOne Programming by Implicit Enumeration," Vol. 19, No. 3, Sep. 1972,
pp. 435447.
ROSE, M., "Determination of the Optimal Investment in End Products and Repair Resources," Vol. 20, No. 1, Mar. 1973, pp.
147159.
ROSENBERG, L. and M. Klein, "Deterioration of Inventory and Equipment," Vol. 7, No. 1, Mar. 1960, pp. 4962.
ROSENBLATT, D., "On the Graphs and Asymptotic Forms of Finite Boolean Relation Matrices and Stochastic Matrices," Vol. 4,
No. 2, June 1957, pp. 151167.
ROSENBLATT, H. and W. Wolman, "The New Military Standard 414 for Acceptance Inspection by Variables," Vol. 6, No. 2,
June 1959, pp. 173182.
ROSENBLATT, J., "Statistical Aspects of Collision Warning System Design Under the Assumption of Constant Velocity
Courses," Vol. 8, No. 4, Dec. 1961. pp. 317341.
CUMULATIVE INDEX FOR VOLUMES 120 833
ROSHOLDT, E. F. and R. B. Hunt, "Determining Merchant Shipping Requirements in Integrated Military Planning," Vol. 7,
No. 4, Dec. 1960, pp. 545575.
ROSHOLDT, E. F. and R. B. Hunt, "The Concepts of Notional Ship and Notional Value in Logistics Capability Studies Involving
Merchant Ships," Vol. 7, No. 1, Mar. 1960. pp. 16.
ROSHOLDT, E. F., R. W. Lewis, and W. L. Wilkinson, "A MultiMode Transportation Network Model," Vol. 12, Nos. 3 & 4,
Sept.Dec. 1965, pp. 261274.
ROSS, S. M., C. Derman and G. J. Lieberman, "On Optimal Assembly of Systems," Vol. 19, No. 4, Dec. 1972, pp. 569574.
ROSSER, J. B., "The Probability of Survival of a Subterranean Target Under Intensive Attack," Vol. 14, No. 4, Dec. 1967, pp.
435451.
ROTOLO, L. S. and R. S. Ledley, "A Heuristic Concept and an Automatic Computer Program Aid for Operational Simulation,"
Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 231244.
RUEHLOW, S. E. , "The Interdependency of Logistic and Strategic Planning." Vol. 1 , No. 4, Dec. 1954, pp. 237257.
RUTEMILLER, H. C. and G. G. Brown, "A Cost Analysis of Sampling Inspection under Military Standard 105D," Vol. 20, No. 1,
Mar. 1973, pp. 181199.
SAATY, T. L., "Optimum Positions for m Airports." Vol. 19, No. 1, Mar. 1972, pp. 101109.
SAATY,T. L., "On Nonlinear Optimization in Integers," Vol. 15, No. l.Mar. 1968, pp. 122.
SAATY, T. L., "Seven More Years of Queues," Vol. 13, No. 4, December 1966, pp. 447476.
SAATY, T. and S. Gass, "The Computational Algorithm for the Parametric Objective Function," Vol. 2, Nos. 1 & 2, Mar.June
1955, pp. 3945.
SACHAKLIAN. H. A., "Risk and Hazard in Logistics Planning," Vol. 2, No. 4, Dec. 1955, pp. 217224.
SACKS, J. and C. Derman. "Replacement of Periodically Inspected Equipment," Vol. 7, No. 4, Dec. 1960. pp. 597607.
SAHNEY, V. K. , "Scheduling Data Transmission under an {S ,; O } Policy," Vol. 19, No. 4, Dec. 1972, pp. 725735.
ST. JOHN, L. R., "Trends in Logistics," Vol. 1, No. 3, Sept. 1954, pp. 182190.
SAIPE, A. L. and A. A. Cunningham. "Heuristic Solution to a Discrete Collection Model," Vol. 19, No. 2, June 1972. pp. 379388.
SAKAGUCHI, M., "Pure Strategy Solutions to Blotto Games in Closed Auction Bidding," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962,
pp. 253263.
SALEH, A. K. MD. E. and M. Ahsanullah, "Optimum Allocation of Quantiles in Disjoint Intervals for the Blues of the Parameters
of Exponential Distribution when the Sample is Censored in the Middle," Vol. 17, No. 3, Sept. 1970, pp. 331349.
SALKIN, H. M. and P. Breining, "Integer Points on the Gomory Fractional Cut (Hyperplane)," Vol. 18, No. 4, Dec. 1971, pp.
491496.
SALKIN, H. M., "A Note Comparing Glover's and Young's Simplified Primal Algorithms," Vol. 19, No. 2, June 1972. pp. 399
402.
SALTZMAN.S. and J. Delfausse, "Values for Optimum Reject Allowances," Vol. 13, No. 2, June 1966, pp. 147157.
SALVESON, M. E., "Principles of Dynamic Weapon Systems Programming," Vol. 8, No. 1 , Mar. 1961 , pp. 79110.
SAPOSNIK, R., A. R. Lindeman and V. L. Smith, "Allocation of a Resource to Alternative Probabilistic Demands: Transport
Equipment Pool Assignments," Vol. 6, No. 3, Sept. 1959, pp. 193207.
SAVAGE, I. R., "Cycling," Vol. 3, No. 3, Sept. 1956, pp. 163177.
SAVAGE, I. R., "Surveillance Problems," Vol. 9, Nos. 3 & 4, Sept.Dec. 1962, pp. 187209.
SAVAGE, I. R., "Surveillance Problems: Poisson Models with Noise," Vol. 11, No. 1, Mar. 1964, pp. 113.
SAVAGE, I. R. and G. Antelman, "Characteristic Functions of Stochastic Integrals and Reliability Theory," Vol. 12, Nos. 3 & 4,
Sept.Dec. 1965, pp. 199222.
SAVAGE, I. R. and G. Antelman, "Surveillance Problems: Wiener Processes," Vol. 12, No. 1, Mar. 1965, pp. 3556.
SCARF, H. E., "Some Remarks on Bayes Solutions to the Inventory Problem," Vol. 7, No. 4, Dec. 1960, pp. 591596.
SCHAFER, R. E., "Bayes Single Sampling Plans by Attributes Based on the Posterior Risks." Vol. 14, No. 1, Mar. 1967, pp.
8188.
SCHAFER, R. E. and N. D. Singpurwalla, "A Sequential Bayes Procedure for Reliability Demonstration," Vol. 17, No. 1, Mar.
1970, pp. 5567.
SCHERER, F. M., "A Note on TimeCost Tradeoffs in Uncertain Empirical Research Projects," Vol. 13. No. 3, Sept. 1966, pp.
349350.
SCHERER, F. M., "TimeCost Tradeoffs in Uncertain Empirical Research Projects," Vol. 13, No. 1, March 1966, pp. 7182.
SCHMIDT, J. W., JR. and W. E. Biles, "A Note on a Paper by Houston and Huffman," Vol. 19, No. 3, Sept. 1972, pp. 561567.
SCHOEFFLER, M. S., W. V. Caldwell, C. H. Coombs, and R. M. Thrall, "A Model for Evaluating the Output of Intelligence
Systems," Vol. 8, No. 1 , Mar. 1961 , pp. 2540.
SCHRADY,D. A., "Operational Definitions of Inventory Record Accuracy," Vol. 17. No. l,Mar. 1970. pp. 133142.
SCHRADY, D. A. and U. C. Choe, "Models for MultiItem Continuous Review Inventory Policies Subject to Constraints," Vol.
18, No. 4, Dec. 1971, pp. 451463.
SCHRADY, D. A., "A Deterministic Inventory Model for Reparable Items," Vol. 14, No. 3, Sept. 1967, pp. 391398.
SCHRAGE, L., "Using Decomposition in Integer Programming," Vol. 20, No. 3, Sept. 1973, pp. 469476.
SCHROEDER, R. G., "On Some Stochastic Tactical Antisubmarine Games," Vol. 14, No. 3, Sept. 1967, pp. 291311.
SCHWARTZ, A. N. and A. J. Boness, "Aircraft Replacement Policies in the Naval Advanced Jet Pilot Training Program: A
Practical Example of DecisionMaking Under Incomplete Information," Vol. 16, No. 2, June 1969, pp. 237257.
834 CUMULATIVE INDEX FOR VOLUMES 120
SCHWARTZ, A. N., J. A. Sheler and C. R. Cooper, "Dynamic Programming Approach to the Optimization of Naval Aircraft
Rework and Replacement Policies," Vol. 18, No. 3, Sept. 1971, pp. 395414.
SCOFIELD, E. K., "Research for Command Logistics," Vol. 7, No. 4, Dec. 1960, pp. 315333.
SCOTT, M., "A Queueing Process with Varying Degree of Service," Vol. 17, No. 4, Dec. 1970, pp. 515523.
SEGEL, F. W., S. E. Haber and H. Solomon, "Statistical Auditing of LargeScale Management Information Systems," Vol.
19, No. 3, Sept. 1972, pp. 449459.
SEGEL, F. W., M. Denicoff, J. Fennell, S. E. Haber, W. H. Marlow and H. Solomon, "The Polaris Military Essentiality System,"
Vol. 11, No. 4, Dec. 1964, pp. 235257.
SETHI, S. P. and T. E. Morton, "A Mixed Optimization Technique for the Generalized Machine Replacement Problem," Vol. 19,
No. 3, Sept. 1972, pp. 471481.
SHAPLEY, L. S., "Complements and Substitutes in the Optimal Assignment Problem," Vol. 9, No. 1, Mar. 1962, pp. 4548.
SHAPLEY, L. S., "Equilibrium Points in Games with Vector Payoffs," Vol. 6, No. 1, Mar. 1959, pp. 5761.
SHAPLEY, L. S., "On Balanced Sets and Cores," Vol. 14, No. 4, Dec. 1967, pp. 453460.
SHAPLEY, L. S., "On Network Flow Functions," Vol. 8, No. 2, June 1961, pp. 151158.
SHARPE, W. F., "Aircraft Compartment Design Criteria for the Army Deployment Mission," Vol. 8, No. 4, Dec. 1961, pp.
381394.
SHEDLER, G. S. and D. P. Gaver, "Control Variable Methods in the Simulation of a Model of a Multiprogrammed Computer
System," Vol. 18, No. 4, Dec. 1971, pp. 435450.
SHELER, J. A., A. N. Schwartz and C. R. Cooper, "Dynamic Programming Approach to the Optimization of Naval Aircraft
Rework and Replacement Policies," Vol. 18, No. 3, Sept. 1971, pp. 395414.
SHERBROOKE, C. C, "Discrete Compound Poisson Processes and Tables of the Geometric Poisson Distribution," Vol. 15,
No. 2, June 1968. pp. 189204.
SHERE, K. D. and E. A. Cohen, Jr., "A Defense Allocation Problem with Development Costs," Vol. 19, No. 3, Sept. 1972, pp.
525537.
SHOEMAKER, R. M., "Principles of Logistics A Provisional Definition," Vol. 8, No. 2, June 1961, pp. 159173.
SHUBIK, M., "Toward a Study of Bidding Processes: Part IV — Games with Unknown Costs," Vol. 14, No. 4, Dec. 1967, pp.
415433.
SHUBIK, M. and Greismer, J. H., "Toward a Study of Bidding Processes: Some ConstantSum Games," Vol. 10, No. 1. Mar.
1963, pp. 1121.
SHUBIK, M. and J. H. Greismer, "Toward a Study of Bidding Processes, Part II: Games with Capacity Limitations," Vol. 10,
No. 2, June 1963, pp. 151173.
SHUBIK, M. and J. H. Greismer, "Toward a Study of Bidding Processes, Part III: Some Special Models," Vol. 10, No. 3, Sept.
1963, pp. 199217.
SHUBIK, M. and G. L. Thompson, "Games of Economic Survival," Vol. 6, No. 2, June 1959, pp. 11 1123.
SICILIA, G. T., H. D. Ratliff and S. H. Lubore, "Determining the Most Vital Link in a Flow Network," Vol. 18, No. 4, Dec. 1971,
pp. 497502.
SILVER, E. A., "Three Ways of Obtaining the Average Cost Expression in a Problem Related to Joint Replenishment In
ventory Control." Vol. 20, No. 2, June 1973, pp. 241254.
SILVER, E. A., "Inventory Allocation Among an Assembly and Its Repairable Subassemblies," Vol. 19, No. 2, June 1972. pp.
261280.
SIMON, R. M., "The Reliability of Multicomponent Systems Subject to Cannibalization," Vol. 19, No. l.Mar. 1972, pp. 114.
SIMONNARD, M. A., and G. F. Hadley. "A Simplified TwoPhase Technique for the Simplex Method," Vol. 6, No. 3, Sept.
1959, pp. 221226.
SIMONNARD, M. A. and G. F. Hadley, "Maximum Number of Iterations in the Transportation Problem," Vol. 6, No. 2, June
1959, pp. 125129.
SIMMONS, K. W. and J. Bracken, "Minimizing Reductions in Readiness Caused by Time Phased Decreases in Aircraft Over
haul and Repair Activities," Vol. 13, No. 2, June 1966, pp. 159165.
SIMPSON, J. R., "A Formula for Decisions on Retention or Disposal of Excess Stock," Vol. 2, No. 3, Sept. 1955, pp. 145155.
SINGPURWALLA, N. D. and R. E. Schafer, "A Sequential Bayes Procedure for Reliability Demonstration," Vol. 17, No. 1,
Mar. 1970, pp. 5567.
SINGPURWALLA, N. D. and C. M. Harris, "On Estimation in Weibull Distributions with Random Scale Parameters, Vol. 16,
No. 3, Sept. 1969, pp. 405410.
SITGREAVES, R., S. Haber and H. Solomon, "A Demand Prediction Technique for Items in Military Inventory Systems,"
Vol. 16, No. 3, Sept. 1969, pp. 297308.
SITGREAVES, R. and S. E. Haber, "A Unified Model for Demand Prediction in the Context of Provisioning and Replenishment,"
Vol. 19, No. 1, Mar. 1972, pp. 2942.
SITGREAVES, R. and S. E. Haber, "A Methodology for Estimating Expected Usage of Repair Parts with Application to Parts
with No Usage History," Vol. 17, No. 4, Dec. 1970, pp. 535546.
SMITH, D. E. , "Requirements of an 'Optimizer' for Computer Simulations," Vol. 20. No. 1 , Mar. 1973, pp. 161179.
SMITH, J. W., "A Plan to Allocate and Procure Electronic Sets by the Use of Linear Programming Techniques and Analytical
Methods of Assigning Values to Qualitative Factors," Vol. 3, No. 3, Sept. 1956, pp. 151162.
CUMULATIVE INDEX FOR VOLUMES 120 335
SMITH, M. W. and J. E. Walsh, "Optimum Sequential Search with Discrete Locations and Random Acceptance Errors," Vol.
18, No. 2, June 1971 , pp. 159167.
Smith, R. A., J. E. Cremeans, and G. R. Tyndall, "Optimal Multicommodity Network Flows with Resource Allocation," Vol.
17, No. 3, Sept. 1970, pp. 269279.
SMITH, V. L., R. Saposnik, and A. R. Lindeman, "Allocation of a Resource to Alternative Probabilistic Demands: Transport
Equipment Pool Assignments," Vol. 6, No. 3, Sept. 1959, pp. 193207.
SMITH, W. E., "Various Optimizers for SingleStage Production," Vol. 3, Nos. 1 & 2, Mar.June 1956, pp. 5966.
SODARO, D., "MultiProduct Production Scheduling Via Extreme Point Properties of Linear Programming," Vol. 15, No. 2,
June 1968, pp. 287300.
SOKOLOWSKY, D., A. J. Hoffman, and J. W. Gaddum, "On the Solution of the Caterer Problem," Vol. 1, No. 3, Sept. 1954,
pp. 223229.
SOLAND, R. M. and J. Bracken, "Statistical Decision Analysis of Stochastic Linear Programming Problems," Vol. 13, No. 3,
Sept. 1966, pp. 205225.
SOLAND, R. M., "An Algorithm for Separable Piecewise Convex Programming Problems," Vol. 20, No. 2, June 1973, pp.
325340.
SOLAND, R. M. and D. Gross, "A Branch and Bound Algorithm for Allocation Problems in Which Constraint Coefficients
Depend upon Decision Variables," Vol. 16, No. 2, June 1969, pp. 157174.
SOLOMON, H., "A Note on a First Application of Clustering Procedures to Fleet Material Condition Measurements," Vol. 18,
No. 3, Sept. 1971, pp. 415421.
SOLOMON, H., S. Haber, and R. Sitgreaves, "A Demand Prediction Technique for Items in Military Inventory Systems,"
Vol. 16, No. 3, Sept. 1969, pp. 297308.
SOLOMON, H., "The Polaris Military Essentiality System," Vol. 11, No. 4, Dec. 1964, pp. 235257.
SOLOMON, H., "The Determination and Use of MilitaryWorth Measurements for Inventory Systems," Vol. 7, No. 4, Dec.
1960, pp. 529532.
SOLOMON, H. and M. Denicoff, "Simulations of Alternative Allowance List Policies," Vol. 7, No. 2, June 1960, pp. 137149.
SOLOMON, H., M. Denicoff, and J. P. Fennell, "Summary of a Method for Determining the Military Worth of Spare Parts,"
Vol. 7, No. 3, Sept. 1960, pp. 221234.
SOLOMON, H., M. Denicoff, J. Fennell, S. E. Haber, and W. H. Marlow, "A Polaris Logistics Model," Vol. 11, No. 4, Dec. 1964,
pp. 259272.
SOLOMON, H., M. Denicoff, J. Fennell, S. E. Haber, W. H. Marlow, and F. W. Segel, "The Polaris Military Essentiality
System," Vol. 11, No. 4, Dec. 1964, pp. 235257.
SOLOMON, H., S. E. Haber, and F. W. Segel, "Statistical Auditing of LargeScale Management Information Systems," Vol. 19,
No. 3, Sept. 1972, pp. 449459.
SOLOMON, M. J. , "A Scientific Method for Establishing Reorder Points," Vol. 1 , No. 4, Dec. 1954, pp. 289294.
SORIANO, A. and D. Gross, "On the Economic Application of Airlift to Product Distribution and Its Impact on Inventory Levels,"
Vol. 19, No. 3, Sept. 1972, pp. 501507.
SOYSTER, A. L., "MultiProduct Production Scheduling Via Extreme Point Properties of Linear Programming," Vol. 15,
No. 2, June 1968, pp. 287300.
SOYSTER, H. R., "Mathematical Programming and Evaluation of Freight Shipment Systems, Application, and Analysis,"
Part I, "Applications," Vol. 4, No. 3, Sept. 1957, pp. 237242.
SPINNER, A. H., "Sequencing Theory Development to Date," Vol. 15, No. 2, June 1968, pp. 319330.
SPIVEY, W. A. and H. Tamura, "Goal Programming in Econometrics," Vol. 17, No. 2, June 1970. pp. 183192.
SRINIVASAN, V., "A Hybrid Algorithm for the One Machine Sequencing Problem to Minimize Total Tardiness," Vol. 18,
No. 3, Sept. 1971, pp. 317327.
SRINIVASAN, V. and G. L. Thompson, "An Operator Theory of Parametric Programming for the Transportation Problem — I,"
Vol. 19, No. 2, June 1972, pp. 205225.
SRINIVASAN, V. and G. L. Thompson, "An Operator Theory of Parametric Programming for the Transportation Problem — II,"
Vol. 19, No. 2, June 1972, pp. 227252.
SRINIVASAN, V. and G. L. Thompson, "Determining Optimal Growth Paths in Logistics Operations," Vol. 19, No. 4, Dec.
1972, pp. 575599.
STANLEY, E. D., D. P. Honig, and L. Gainen, "Linear Programming in Bid Evaluation," Vol. 1, No. 1, Mar. 1954, pp. 4954.
STEDRY, A. C. and R. G. Brandenburg, "Toward a MultiStage Information Conversion Model of the Research and Develop
ment Process," Vol. 13, No. 2, June 1966, pp. 129146.
STEDRY, A. C. and H. D. Brecht, "Toward Optimal Bidding Strategies," Vol. 19, No. 3, Sept. 1972, pp. 423434.
STEGER, W. A., M. A. Geisler, and W. W. Haythorn, "Simulation and the Logistics Systems Laboratory," Vol. 10, No. 1, Mar.
1963, pp. 2354.
STEGER, W. A. and J. W. Petersen, "Design Change Impacts on Airframe Parts Inventories," Vol. 5, No. 3, Sept. 1958, pp.
241255.
STEIN, C, JR., "Logistics Research Programs of the U.S. Army, U.S. Air Force, and U.S. Navy, "Briefing on the Logistics
Research Program of the Navy," Vol. 5, No. 3, Sept. 1958, pp. 225229.
STEINBERG, D. I., "The Fixed Charge Problem," Vol. 17, No. 2, June 1970, pp. 217235.
836 CUMULATIVE INDEX FOR VOLUMES 120
STEINHAUS, H., "Definitions for a Theory of Games and Pursuit." Vol. 7, No. 2, June 1960 pp. 105108.
STERNLIGHT, D., "The Fast Deployment Logistic Ship Project: Economic Design and Decision Techniques," Vol. 17, No. 3,
Sept. 1970, pp. 373387.
STONE, L. D., "Incremental Approximation of Optimal Allocations," Vol. 19, No. 1, Mar. 1972, pp. 111122.
STONE, L. D., "Total Optimality of Incrementally Optimal Allocations," Vol. 20, No. 3, Sept. 1973, pp. 419430.
STONE, L. D. and H. R. Richardson, "Operations Analysis During the Underwater Search for Scorpion," Vol. 18, No. 2, June
1971, pp. 141157.
STROLLER, D. S., "Some Queuing Problems in Machine Maintenance." Vol. 5, No. 1, Mar. 1958, pp. 8387.
SUURBALLE, J. W. and K. T. Marshall, "A Note on Cycling in the Simplex Method," Vol. 16, No. 1, Mar. 1969, pp. 121137.
SUZUKI, G., "Bid Evaluation for Procurement of Aviation Fuel at DFSC: A Case History," Vol. 14, No. 1, Mar. 1967, pp. 115129.
SUZUKI, G., "Procurement and Allocation of Naval Electronic Equipments," Vol. 4. No. 1, Mar. 1957, pp. 17.
SWEAT, C. W., "Adaptive Competitive Decision in Repeated Play of a Matrix Game with Uncertain Entries," Vol. 15, No. 3,
Sept. 1968, pp. 425448.
SWEAT, C. W., "A Duel Complicated by False Targets and Uncertainty as to Opponent Type," Vol. 19, No. 2, June 1972, pp.
355367.
SWINSON, G. E. and P. H. Randolph, "The Discrete MaxMin Problem," Vol. 16, No. 3, Sept. 1969, pp. 309314.
SZWARC, W., "Elimination Methods in the m x n Sequencing Problem," Vol. 18. No. 3, Sept. 1971, pp. 295305.
SZWARC, W., "Some Remarks on the Time Transportation Problem," Vol. 18, No. 4, Dec. 1971, pp. 473485.
SZWARC, W., "The Transportation Paradox," Vol. 18, No. 2, June 1971, pp. 185202.
SZWARC, W., "On Some Sequencing Problems," Vol. 15, No. 2, June 1968, pp. 127156.
SZWARC, W., "The Truck Assignment Problem," Vol. 14, No. 4, Dec. 1967, pp. 529557.
TAHA, H. A., "Concave Minimization over a Convex Polyhedron," Vol. 20, No. 3, Sept. 1973, pp. 533548.
TAMURA, H. and W. A. Spivey, "Goal Programming in Econometrics," Vol. 17, No. 2, June 1970, pp. 183192.
TAYLOR, J. G., "On the Isbell and Marlow Fire Programming Problem," Vol. 19, No. 3, Sept. 1972, pp. 539556.
TAYLOR, J. G., "A SquaredVariable Transformation Approach to Nonlinear Programming Optimality Conditions," Vol. 20,
No. 1, Mar. 1973, pp. 2539.
TAYLOR, J. G., "Target Selection in Lanchester Combat: LinearLaw Attrition Process," Vol. 20, No. 4, Dec. 1973, pp. 673697.
TAYLOR, R. J. and S. P. Thompson, "On a Certain Problem in Linear Programming," Vol. 5, No. 2, June 1958, pp. 171187.
TEAGER, H. M., "The Marriage of OnLine Human Decision with Computer Programs," Vol. 7, No. 4, Dec. 1960, pp. 379383.
THOLEN, A. D., J. Bracken, C. B. Brossman, and C. B. Magruder, "A Theater Materiel Model," Vol. 12, Nos. 3 & 4, Sept.
Dec. 1965, pp. 295313.
THOMAS, C. J. and D. V. Mastran, "Decision Rules for Attacking Targets of Opportunity," Vol. 20, No. 4, Dec. 1973, pp. 661672.
THOMAS, M. E. and M. Grunspan, "Hyperbolic Integer Programming," Vol. 20, No. 2, June 1973, pp. 341356.
THOMPKINS, C, "Some Methods of Computational Attack on Programming Problems Other than the Simplex Method
(Abstract)," Vol. 4, No. 1, Mar. 1957, pp. 9596.
THOMPSON, D. E., "Stochastic Duels Involving Reliability," Vol. 19, No. 1, Mar. 1972, pp. 145148.
THOMPSON, G. L. and O. Morgenstern, "An Open Expanding Economy Model," Vol. 16, No. 4, Dec. 1969, pp. 443457.
THOMPSON, G. L. and Y. Ijiri, "Mathematical Control Theory Solution of an Interactive Accounting Flows Model," Vol. 19,
No. 3, Sept. 1972, pp. 411422.
THOMPSON. G. L. and O. Morgenstern, "A Note on an Open Expanding Economy Model," Vol. 19, No. 3, Sept. 1972, pp.
557559.
THOMPSON, G. L. and V. Srinivasan, "Determining Optimal Growth Paths in Logistics Operations," Vol. 19, No. 4, Dec.
1972, pp. 575599.
THOMPSON, G. L. and V. Srinivasan, "An Operator Theory of Parametric Programming for the Transportation Problem — I,"
Vol. 19, No. 2, June 1972, pp. 205225.
THOMPSON, G. L. and V. Srinivasan, "An Operator Theory of Parametric Programming for the Transportation Problem — II,"
Vol. 19, No. 2, June 1972, pp. 227252.
THOMPSON, G. L., "CPM and DCPM Under Risk," Vol. 15, No. 2, June 1968, pp. 233240.
THOMPSON, G. L., "Decision Making and New Mathematics," Vol. 3, No. 3, Sept. 1956, pp. 141150.
THOMPSON, G. L., "Recent Developments in the JobShop Scheduling Problem," Vol. 7, No. 4, Dec. 1960, pp. 585589.
THOMPSON, G. L., F. K. Levy, and J. D. Wiest, "Multiship, Multishop, WorkloadSmoothing Program," Vol. 9, No. 1, Mar.
1962, pp. 3744.
THOMPSON, G. L., and M. Shubik, "Games of Economic Survival," Vol. 6, No. 2, June 1959, pp. 1 1 1123.
THOMPSON, P. M., "Editing Large Linear Programming Matrices," Vol. 4, No. 1, Mar. 1957, pp. 97100.
THOMPSON, S. P. and R. J. Taylor, "On A Certain Problem in Linear Programming," Vol. 5, No. 2, June 1958, pp. 171187.
THOMPSON, S. P. and A. J. Ziffer, "The Watchdog and the Burglar." Vol. 6, No. 2, June 1959, pp. 165172.
THRALL, R. M., "A Note on Incentive Fee Contracting," Vol. 12, Nos. 3 & 4, Sept.Dec. 1965, pp. 331333.
THRALL, R. M. and D. R. Howes, "A Theory of Ideal Linear Weights for Heterogenous Combat Forces," Vol. 20, No. 4, Dec.
1973, pp. 645659.
THRALL, R. M., W. V. Caldwell, C. H. Coombs, and M. S. Schoeffler, "A Model for Evaluating the Output of Intelligence
Systems," Vol. 8, No. 1, Mar. 1961, pp. 2540.
CUMULATIVE INDEX FOR VOLUMES 120 837
THRALL, R. M., C. H. Coombs, and W. Caldwell, "Linear Model for Evaluating Complex Systems," Vol. 5, No. 4, Dec. 1958.
pp. 347361.
THRALL, R. M. and W. F. Lucas, "nPerson Games in Partition Function Form," Vol. 10, No. 4, Dec. 1963, pp. 281298.
TINDALL, J. B. and J. J. Jarvis, "Minimal Disconnecting Sets in Directed MultiCommodity Networks," Vol. 19, No. 4, Dec.
1972, pp. 681690.
TIPLITZ.C. I., "Convergence of the Bounded Fixed Charge Programming Problem," Vol. 20, No. 2, June 1973, pp. 367375.
TOMPKINS, C. B., "Some Aspects of Mathematics in Social Sciences," Vol. 7, No. 4, Dec. 1960. pp. 335356.
TOWNSLEY, R. J. and W. Candler, "Quadratic as Parametric Linear Programming," Vol. 19, No. 1 , Mar. 1972, pp. 183189.
TREPP, R. C. and V. J. Presutti, Jr., "More Ado About Economic Order Quantities (EOQ)," Vol. 17, No. 2, June 1970, pp. 243
251.
TURNER, W. C, P. M. Chare, D. C. Montgomery, "Optimal Interdiction Policy for a Flow Network," Vol. 18, No. 1, Mar.
1971, pp. 3745.
TYNDALL, G. R., J. E. Cremeans, and R. A. Smith, "Optimal Multicommodity Network Flows with Resource Allocation,"
Vol. 17, No. 3, Sept. 1970, pp. 269279.
UZAWA, H., K. J. Arrow, and L. Hurwicz, "Constraint Qualifications in Maximization Problems," Vol. 8, No. 2, June 1961,
pp. 175191.
VACHANI, M., "Determining Optimum Reject Allowances for Deteriorating Production Systems," Vol. 16, No. 3, Sept. 1969,
pp. 275286.
VAN DE PANNE, C. and A. Whinston, "Simplicial Methods for Quadratic Programming," Vol. 11, No. 4, Dec. 1964, pp. 273
302.
VARLEY, T. C. and J. Bracken, "A Model for Determining Protection Levels for Equipment Classes within a Set of Subsystems,"
Vol. 10, No. 3, Sept. 1963, pp. 257262.
VAZSONYI, A. and W. Karush, "Mathematical Programming and Employment Scheduling," Vol. 4, No. 4, Dec. 1957, pp. 297
320.
VEINOTT, A. F., Jr. and S. A. Bessler, "Optimal Policy for a Dynamic MultiEchelon Inventory Model," Vol. 13, No. 4, Dec.
1966, pp. 355389.
VERGIN, R. C, "Optimal Renewal Policies for Complex Systems," Vol. 15, No. 4, Dec. 1968, pp. 523534.
VERHULST, M., "The Concept of a 'Mission,' " Vol. 3, Nos. 1 & 2, Mar. June 1956, pp. 4557.
VON LANZENAUER. C. H., "Production and Employment Scheduling in Multistage Production Systems," Vol. 17, No. 2,
June 1970, pp. 193198.
VON NEUMANN, J., "A Numerical Method to Determine Optimum Strategy," Vol. 1, No. 2, June 1954, pp. 1091 15.
WADSWORTH, G. P., J. G. Bryan, and T. M. Whitin, "A MultiStage Inventory Model," Vol. 2, Nos. 1 & 2, Mar.June 1955,
pp. 25—37.
WAGGENER, H. A., "Bid Evaluation for Procurement of Aviation Fuel at DFSC: A Case History," Vol. 14, No. 1, Mar. 1967,
pp. 115129.
WAGNER, H. M., "A Postscript to 'Dynamic Problems in the Theory of the Firm,' " Vol. 7, No. 1, Mar. 1960, pp. 712.
WAGNER, H. M., "An Integer LinearProgramming Model for Machine Scheduling," Vol. 6, No. 2, June 1959, pp. 131140.
WAGNER, H. M., "The Dual Simplex Algorithm for Bonded Variables," Vol. 5, No. 3, Sept. 1958, pp. 257261.
WAGNER, H. M., "The Lower Bounded and Partial Upper Bounded Distribution Model," Vol. 20, No. 2, June 1973, pp. 265268.
WAGNER, H. M., and T. M. Whitin, "Dynamic Problems in the Theory of the Firm," Vol. 5, No. 1, Mar. 1958, pp. 5374.
WALSH, J. E., "A General Simulation Model for Logistics Operation in a Randomly Damaged System," Vol. 7, No. 4, Dec.
1960, pp. 453470.
WALSH, J. E. and I. M. Garfunkel, "A Method for FirstStage Evaluation of Complex ManMachine Systems," Vol. 7, No. 1,
Mar. 1960, pp. 1319.
WALSH, J. E. and M. Luther, "A Difficulty in Linear Programming for Transportation Problems," Vol. 5, No. 4, Dec. 1958, pp.
363366.
WALSH, J. E. and M. W. Smith, "Optimum Sequential Search with Discrete Locations and Random Acceptance Errors,"
Vol. 18, No. 2, June 1971, pp. 159167.
WEBB, S., "Interactions between the Experiment Designer and the Computer," Vol. 16, No. 3, Sept. 1969, pp. 423^133.
WEIDMAN, D. R., "Optimal Scheduling of Objects in Circulating Systems," Vol. 14, No. 4, Dec. 1967, pp. 559568.
WEIGEL, H. S. and J. E. Cremeans, "The Multicommodity Network Flow Model Revised to Include Vehicle Per Time Period
and Node Constraints," Vol. 19, No. 1, Mar. 1972, pp. 7789.
WEINBERG, R. S. and J. W. Higgins, "The Feasibility of a Global Airlift," Vol. 6, No. 2, June 1959, pp. 891 10.
WEINSTOCK, J. K. and R. R. O'Neill, "The CargoHandling System," Vol. 1, No. 4, Dec. 1954, pp. 282288.
WEISS, G., "On the Theory of Replacement of Machinery with a Random Failure Time," Vol. 3, No. 4, Dec. 1956, pp. 279293.
WEISS, L., "Approximating Maximum Likelihood Estimators Based on Bounded Random Variables," Vol. 15, No. 2, June 1968,
pp. 169178.
WEISS, L., "Confidence Intervals of Preassigned Length for Qualities of Unimodal Populations," Vol. 7, No. 3, Sept. 1960,
pp. 251256.
WEISS, L., " 'Hedging* on Statistical Assumptions," Vol. 8, No. 3, Sept. 1961, pp. 207213.
838 CUMULATIVE INDEX FOR VOLUMES 120
WEISS, L., "On Estimating Location and Scale Parameters from Truncated Samples," Vol. 11, Nos. 2 and 3, JuneSept. 1964,
pp. 125134.
WEISS, L., "On the Asymptotic Distribution of an Estimate of a Scale Parameter," Vol. 10, No. 1, Mar. 1963, pp. 19.
WEISS, L., "On the Estimation of Scale Parameters," Vol. 8, No. 3, Sept. 1961, pp. 245256.
WEISS, L., "Asymptotic Inference about a Density Function at an End of Its Range," Vol. 18, No. 1, Mar. 1971, pp. 111114.
WENTLING, L. G., "Programming the Procurement of Airlift and Sealift Forces: A Linear Programming Model for Analysis
of the LeastCost Mix of Strategic Deployment Systems," Vol. 14, No. 2, June 1967, pp. 241255.
WESOLOWSKY, G. O. and R. F. Love, "Location of Facilities with Rectangular Distances Among Point and Area Destinations,"
Vol. 18, No. 1, Mar. 1971, pp. 8390.
WHEELER, A. C, "Stationary (s, S) Policies for a Finite Horison," Vol. 19, No. 4, Dec. 1972, pp. 601619.
WHELAN, D. W., "Material Logistic Support of Weapons Systems," Vol. 8, No. 4, Dec. 1961, pp. 361375.
WHINSTON, A. and C. Van De Panne, "Simplicial Methods for Quadratic Programming," Vol. 11, No. 4, Dec. 1964, pp. 273302.
WHINSTON, A., "Conjugate Functions and Dual Programs," Vol. 12, Nos. 3 & 4, Sept. Dec. 1965, pp. 315322.
WHINSTON, A., "The Bounded Variable Problem — An Application of the Dual Method for Quadratic Programming," Vol.
12, No. 2, June 1965, pp. 173180.
WHITE, T. M. and J. W. T. Youngs, "A Method for Calculating Optimal Inventory Levels and Delivery Time," Vol. 2, No. 3,
Sept. 1955, pp. 157173.
WHITIN, T. M., "On the Span of Central Direction," Vol. 1, No. 1, Mar. 1954, pp. 2535.
WHITIN, T. M., J. G. Bryan, and G. P. Wadsworth, "A MultiStage Inventory Model," Vol. 2, Nos. 1 & 2, Mar.June 1955,
pp. 25—37.
WHITIN, T. M. and G. Hadley, "A Model for Procurement, Allocation, and Redistribution for Low Demand Items," Vol. 8,
No. 4, Dec. 1961, pp. 395414.
WHITIN, T. M. and G. Hadley, "Budget Constraints in Logistics Models," Vol. 8, No. 3, Sept. 1961, pp. 215220.
WHITIN, T. M. and H. M. Wagner, "Dynamic Problems in the Theory of the Firm," Vol. 5, No. 1, Mar. 1958, pp. 5374.
WHITON, J. C, "Programming the Procurement of Airlift and Sealift Forces: A Linear Programming Model for Analysis of the
LeastCost Mix of Strategic Deployment Systems," Vol. 14, No. 2^June 1967, pp. 241255.
WHITON, J. C, "Some Constraints on Shipping in Linear Programming Models," Vol. 14, No. 2, June 1967, pp. 257260.
WICKE, H. H. and J. K. Hale, "An Application of Game Theory to Special Weapons Evaluation," Vol. 4, No. 4, Dec. 1957, pp.
347356.
WIEST, J. D., F. K. Levy, and G. L. Thompson, "Multiship, Multishop, WorkloadSmoothing Program," Vol. 9, No. 1, Mar.
1962, pp. 3744.
WILKINSON, W. L., Min/Max Bounds for Dynamic Network Flows," Vol. 20, No. 3, Sept. 1973, pp. 505516.
WILKINSON, W. L., R. W. Lewis, and E. F. Rosholdt, "A MultiMode Transportation Network Model," Vol. 12, Nos. 3 & 4,
Sept.Dec. 1965, pp. 261274.
WILLIAMS, R. E., Jr., "Some Thoughts on Logistic Planning Factors," Vol. 1, No. 3, Sept. 1954, pp. 173181.
WILSON, A. H. and W. R. Finn, "Improvise or Plan?," Vol. 4, No. 4, Dec. 1957, pp. 263267.
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WOLFE, H. B. and R. F. Meyer, "The Organization and Operation of a Taxi Fleet," Vol. 8, No. 2, June 1961, pp. 137150.
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WOLFOWITZ, J. and J. Kiefer, "Sequential Tests of Hypotheses About the Mean Occurrence Time of a Continuous Param
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WOLLMER, R. D., "Interception in a Network," Vol. 17, No. 2, June 1970, pp. 207216.
WOLLMER, R. D. and J. L. Midler, "Stochastic Programming Models for Scheduling Airlift Operations," Vol. 16, No. 3, Sept.
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WOLMAN, W. and H. Rosenblatt, "The New Military Standard 414 for Acceptance Inspection by Variables," Vol. 6, No. 2, June
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CUMULATIVE INDEX FOR VOLUMES 120
839
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U.S. GOVERNAAENT PRINTING OFFICE: 1973— 541 388:2
\\
L
•
INFORMATION FOR CONTRIBUTORS
The NAVAL RESEARCH LOGISTICS QUARTERLY is devoted to the dissemination of
scientific information in logistics and will publish research and expository papers, including those
in certain areas of mathematics, statistics, and economics, relevant to the overall effort to improve
the efficiency and effectiveness of logistics operations.
Manuscripts and other items for publication should be sent to The Managing Editor, NAVAL
RESEARCH LOGISTICS QUARTERLY, Office of Naval Research, Arlington, Va. 222 1 7 .
Each manuscript which is considered to be suitable material for the QUARTERLY is sent to one
or more referees.
Manuscripts submitted for publication should be typewritten, doublespaced, and the author
should retain a copy. Refereeing may be expedited if an extra copy of the manuscript is submitted
with the original.
A short abstract (not over 400 words) should accompany each manuscript. This will appear
at the head of the published paper in the QUARTERLY.
There is no authorization for compensation to authors for papers which have been accepted
for publication. Authors will receive 250 reprints of their published papers.
Readers are invited to submit to the Managing Editor items of general interest in the held
of logistics, for possible publication in the NEWS AND MEMORANDA or NOTES sections
of the QUARTERLY.
NAVAL RESEARCH
LOGISTICS
QUARTERLY
DECEMBER 1973
VOL. 20, NO. 4
NAVSO P1278
CONTENTS
ARTICLES
Generalized Multicomponent Systems under Cannibalization
A Bayesian Approach to Demand Estimation and Inventory
Provisioning
Readiness and the Optimal Redeployment of Resources
On MaxMin Problems
A Theory of Ideal Linear Weights for Heterogeneous Com
bat Forces
Decision Rules for Attacking Targets of Opportunity
Target Selection in Lanchester Combat: LinearLaw Attri
tion Process
An NStep, 2Variable Search Algorithm for the Compo
nent Placement Problem
Parametric Linear Programming: Some Special Cases
Sequential Search of an Optimal Dosage: NonBayesian
Methods
Further Light on Nonparametric Selection Efficiency
Simplified Estimates of the Parameters of the Double
Exponential Distribution Based on Optimum Order
Statistics from a MiddleCensored Sample
The Single Server Queue in Discrete TimeNumerical
Analysis IV
Alternate Methods of Project Scheduling with Limited
Resources
Optimum Adjustment Policy for a Product with Two
Quality Characteristics
Scheduling with Parallel Processors and Linear Delay Costs
News and Memoranda
Index, Volume 20
Cumulative Index, Volumes 120
OFFICE OF NAVAL RESEARCH
Arlington, Va. 22217
M. HOCHBERG 585
G. F. BROWN, JR., 607
W. F. ROGERS
S. KAPLAN 625
K. C. KAPUR 639
D. R. HOWES, 645
R. M. THRALL
D. V. MASTRAN, 661
C. J. THOMAS
J. G. TAYLOR 673
C. H. HEIDER 699
W. DENT, 725
R. JAGANNATHAN,
M. R. RAO
B. H. EICHHORN 729
E. J. DUDEWICZ, 737
C. FAN
M. AHSANULLAH, 745
M. A. RAHIM
D. HEIMANN, 753
M. F. NEUTS
J. H. PATTERSON 767
W.J. KENNEDY, JR., 785
P. M. GHARE
K. R. BAKER,
A. G. MERTEN
793
805
807
813